It's possible that what you have is an illegally imported counterfit. We seized about two dozen looking very similar to what you show in your pics almost a year ago now with a couple different manufacturer logos. Not saying that yours is for a fact an illegally imported counterfit, and not trying to imply you or the person you bought it from did anything wrong, but if I were you i'd not keep it if I could not verify the pedigree as legal if I even had a hint that it might be an illegal import.
If you hold on to it, and if it is an illegal import, and you have any reason to suspect (which obviously you do from your writings) it might not be a legal, you can be held legally responsible. To my knowledge S&W has not authorized anyone to "copy" their weapons (if i'm wrong on this someone correct me please).
Your best bet is to confirm the pedigree of the weapon first. If S&W has no record of the S/N (and i'll bet they don't), if your not going to return it to S&W i'd recommend that you contact the BATF, explain the situation, and seek help in determing the pedigree if it can be determined, but i'll tell you right now if it is a counterfit they will most likely confiscate the weapon but thats better then you being caught with it in the future and going to jail over it.
There are any number of sceinarios, i've seen all sorts of weird things.
It could be the person you got it from got hold of some countefit lowers and put it together for sale. This is not illegal if the person is licensened or registered to manufacturer a firearm, but using the S&W logo would make it a countefit if they were not licensened to use the logo. Its also possible that you simply have a mis-marked rejected parts version where someone has gotten hold of rejected parts from various S&W suppliers (they should have been destroyed). Its not that the parts are really bad, they could have just been mis-marked and that was the reason they were rejected. In which case the weapon was probably illegally manufactured anyway because the use of the S&W logo implies it was manufactured by S&W and it does not appear to be of S&W production.
It could be, and this is one of those "slim and none" chances, that it is a very early S&W production for the model, and markings changed later, that was not intended to make it to market in which case S&W may or may not have a record of the serial number (the weapon should have been destroyed) and you said they didn't. This is the most unlikely sceinario as i'm sure that S&W does not sell weapons "out the back door" for non-marketed weapons. Its possible that the weapon was an S&W manufacturer and the S/N failed to get recorded some way or another (its happened by accident for other manufacturers), sending it back to S&W would allow them to verify it as one of theirs and properly record the S/N, and they would probably swap it out for you with a current production model.
It could also be a one time thing where S&W did make the weapon and placed it in the market and the weapon is a one-of-a-kind thing that was mismarked, in which case it could end up being a collectors item worth something. However, you would think that S&W would have a record of the S/N. I know at one point the M&P 15 was outsourced for manufacture/shipping/sales so a mis-marking could have happened and its possible in that case a S/N was not recorded at S&W but at the outsourced manufacturer instead, so sending it to S&W would help verify this.
Another thing is that it is a S&W weapon but it was a special unique manufacturer for a specific special client and the S/N does not show up in the normal S/N database. I know that a couple of weapon manufacturers maintain seperate databases for S/N's for specific special client products that are not available to the public. Not sure if S&W does this or not.
Now, the weapon uses the S&W trademark logo, and they tell you there is no record of the weapon, if they did not manufacture the weapon then legally the weapon does not exist. However, you do not own the weapon either regardless of what you paid for it because you can not legally own counterfit unlicensened items, and since the S&W logo appears on the weapon S&W is actually the legal owner of the weapon by presence of their trademark logo because they own the trademark logo. So they can demand that you return the weapon to them, at your expense (but they might work out something with you), as its legally their property if its countefit and uses their logo.
If S&W is willing to take responsibility for the weapon and swap it out for you, and will put that in writing (unlikely) or verify for you thats what is going to happen, then you have nothing to worry about by sending it to them and you end up with a new weapon of which your sure of the pedigree and a warranty. However, if they want it back and will not swap it out then its most likely a counterfit and if you don't send it to them they will simply contact the BATF (or Customs) who will end up seizing the weapon anyway and "possibly" (probably actually as they take a very dim view on illegal firearms) arresting and prosecuting you. The BATF (or Customs) will find you if they want, there are many methods but they will most likely subponea the S&W forums for your IP address and use that to find your ISP then subponea the ISP for the internet access account holder name which will lead them to you. If it is a counterfit or has been imported or manufactured unlawfully, you don't have any terms and no one is under any obligations to make arrangements to accommodate you. It is your legal obligation to verify and ensure, without a doubt, the weapon you posess is legal. This means that if you have any doubt as to the authenticity what so ever, you are legally required to verify the legality of the firearm before retaining posession of it, and if it is not a legal firearm you have a legal obligation to surrender the firearm to a legal entity such as law enforcement....OR.... the owner of the trademarked items used in a counterfit which in this case is S&W (it is their trademarked logo), so no, S&W does not need to meet your terms. If it is counterfit or has been imported or manufactured unlawfully, you have a legal obligation to act in such a manner as to abide by the law and not retain such a weapon. If the weapon is counterfit or has been imported or manufactured unlawfully, and you intentionally retain the weapon, you are in posession of an illegal weapon and although the consequences for posessing such vary depending on the circumstances, if you are prepared to deal with the consequences of fines and jail time and never being able to posess another firearm then good luck. I would not depend on the "reputable" gun store owners and gunsmith to decide your future.
Posession of the weapon by its self is not illegal, but posession of the weapon if you have doubts as to its legality is illegal. If your caught with it you will be required to show it was indeed legal and if you can't do that then your out of luck.
If it were me, i'd choose the least injurious method here and send it to S&W and let them authenticate it. That way the reasonable responsible thing was done and removes the penalties for you. If they don't swap it out then your out a weapon and your money (unless you can get the person you bought it from to refund you), but your not going to risk jail time or fines or attorney fees which will cost you a lot more.
Oh, something else. Look at the bolt. Are the letters "L" and "MP" stamped on the bolt? If so then its an LMT manufacture outsource and they were bad for not sending the S/N over to S&W at one point. I know also that at a few different points the letter prefix for the M&P 15 serial number changed - it was "SM xxxxxx" then it changed to "SW xxxxxx" (the SW did not have a '/' in it like yours does which also makes me think its possibly a counterfit), and now its just "SN xxxxxx" (Its SN on the one I got - no '/'). Try getting S&W to check the serial number without the '/' and see what comes up, i'll bet they don't have a record of that either (which is why its a generally safe bet to guess right off that S&W did not manufacture the weapon because the M&P series did not "usually" have a '/' in the letter prefix for the serial number and if it was outsourced then it could have been mis-marked that way.) Its also possible that a company commissioned S&W to produce the weapon for them, in which case it might have a different serial number prefix, for example the "S/W" like yours has - I know a company called TALO Distributors had S&W produce a run of the M&P 15 series for a surplus 5.45 x 39 round and some of those had the "S/W" prefix on the serial numbers with the S&W logo stamped into the lower on the mag well but the other markings would have been in the regular S&W locations like the caliber on the barrel, however, to my knowledge the 5.56/.223 models of the M&P were never made in a "distributor only" run as those are core components of the line of rifles and if it was something like this the serial number would have been recorded at S&W before the weapons were transfered to the special run distributor, but you will need to send it to S&W for them to make sure.
If S&W does authenticate the weapon for you, make sure you get a letter of authenticity from them - it will cost you (I think) about $50.00 to get the letter but get it and hang on to it. The only way they can verifiy the authenticity is for you to return the weapon to them.