Looks like a 38 Safety, 2nd Model, but you did not supply a serial number? The S&W logo did not appear on any models made prior to 1896, except for the hard rubber stocks that started around 1879. The 2nd Model ran from 1887 to 1890. You may or may not have caliber stamping on the barrel as that was another later era stamping on these guns.
It seems that more often than not, the old S&Ws with function issues are easily solved by a complete dis-assemble and cleaning. That revolver would have shipped most likely in early 1890, so it has 127 years of dirt, dried grease, and debris in the works.
There are many good reference books for vintage S&Ws and I would say that best identification reference out there is by Roy Jinks and Robert Neal, Smith & Wesson 1857 to 1945. There are also a couple of good books that detail function & parts, contains schematics, and deals with repairs on old S&Ws and many other guns by David Chicoine. Antique Firearms Assembly/Disassembly & Gunsmithing Guns of the Old West.
I just noticed that the pivot screw appears to be in backwards. The screw head should be on the left and there is a small alignment mark that matches the pin on the right side. The pin head has a slot that allows the head of the pin to look almost flush with the frame when tightened.
The pivot pin is a two-piece affair that passes thru two ears of the frame, two ears of the barrel, and the ejector pawl. The pawl has a radially-loaded spring in the center, but nothing should go boing when pushing out and reversing the pivot pin, as long as you're careful and go slow. You'll see the registration marks when you get the pin reinstalled. Please do keep us advised.
Larry
If it is what I think you are talking about, that is supposed to work the way you describe. There is a retainer on the lower part of that pivoting latch, preventing the cylinder from unscrewing during use. You have to push the front down and hold that retainer up in order to unscrew the cylinder. Don't let the retainer drop during removal or replacement, since it can score the finish.