Cleaning the bore of a .22LR rifle is rarely necessary. Ever.
If you insist glug a bit of CLP down the barrel and drag your Boresnake thru the bore. If you're anal about cleaning drag it thru twice.
The C in CLP is Cleaner. No need for solvents or "gun scrubbers" on the M&P15-22 (nor the M16 Rifle or M4 Carbine either). CLP on a shop rag or toothbrush is all ya to clean the receivers and bolt carrier group. Gun store will be more than glad to sell you a myriad of products that you don't need...
Do NOT remove the fire control parts from the lower receiver unless absolutely necessary. Removing them adds unnecessary wear to the pin holes in the receiver and it's easy to mis-assemble the parts. You should be aware of how to do this, but it's seldom necessary. And if you keep unnecessary "gun scrubbers" out of this area the CLP will do all 3 phases of its job.
-- Chuck