hopefully this rare carbine makes banner. This post keeps getting buried and I had a hard time finding it again. I hope you find the butt plate but there is a chance it was lost way before the carbine was lost. I have dug several pre-1850's musket butt plates that were by themselves that had obviously come off.
If you never find it I would scour the internet relic sites and try and find a loose dug Maynard butt plate and marry it to a board for display. or you could paint the outline of one.
If I found that it would make my year. Not many people can claim they found a Civil War weapon and yours is one of the scarcer variants. As much as I hate to discuss values; it might put it into context for viewers who are not familiar with this sort of thing. I would value this relic at a minimum of $500. it could be worth up to $1000; maybe a little more to the right person and anywhere in between. congratulations an a very historic weapon and relic.
Pulling the exact serial number would be neat; but I would not worry about it too much. your gun was made circa 1863-64 or 65 at the latest. The only real benefit of pulling a number would if there was a known specific issued serial number match. or a 'range match'. A range match might indicate what unit it was originally issued to. but more often than not there are no matches.
so in my opinion it is not worth damaging/screwing up cleaning it. I have seen way too many relics damaged/screwed up over the years. It is desert iron so it should be stable. the fact the cartridge was still in it adds to its value to its historical interest. it is beautiful as it is and will look amazing on your wall. I wish I found it. awesome find!!!!!!!!!!!!!!!!!!!