Be careful of using a sample of 1 for your ammunition model. Bullet sizes varied all over the place in the early cartridge era.
Your oversize bullet groups aren’t that bad for a gun just put back into service. Antiques generally require “shooting in”. Accuracy at first can range from disappointing to down right dismal. Keep shooting and cleaning. After several trips to the range, and a couple hundred rounds, your carbine ought to settle down and start hitting well, even with some pitting in the bore.
If the muzzle is bad, pitted or has lots of cleaning rod wear, try very light loads, like 15-20gr. of FF with some cornmeal filler to get good powder compression. Put a card wad between the bullet and filler. It will probably shoot low at 100 yards, but may group better. DO NOT use the smokeless powder method of a cotton fluff filler between the bullet and powder with black powder. You run a very high risk of ringing your chamber.
My experience is that the cutoff 515141 will work very well when it’s sized a couple thousandths over groove diameter. My New Model carbine is 0.518” groove diameter and I size to 0.520” (both rings). If you swage it way down in sizing, the base can become dished, which will hurt accuracy. Carefully file the bases flat before loading.
Have patience grasshopper.
On your serial number question, when M-1865s came out, they restarted the serial numbers at 1. To add complication, Burnside contract Spencer M-1865 carbines also start with 1. Carbines with numbers between 1 and about 11,000 can be either Spencer or Burnside M-1865s. (M-1860s were rifles in that serial range, but no carbines.) Between sn ~11,000 to ~24,000, you can have a Model 1860 carbine, a Spencer made M-1865 carbine or a Burnside made M-1865 carbine. Between numbers 24K to 34.5K, you could have a Spencer M-1860 or Burnside M-1865, but no Spencer M-1865s. Past 35K to about 65K there are only M-1860s. If the date on the Springfield Research Service hit is before mid 1865, it’s for sure an M-1860 in 56-56.
If it’s after May of '65, it’s anybody’s guess which gun is the one recorded. One scant clue is if it’s in a regular Army unit, it would have been an M-1865 of either manufacturer in 56-50 as the earlier models (even the sleeved 56-50 M-1860s) were considered second class and issued only to militia, teamsters, scouts and so on. Unfortunately, the worst records are from the regular Army, so most hits come from militia units which could have still had M-1860s.
Here's a summary:
M-1860 carbine serial numbers – ~11,000 through ~62,000, with sn 22,000 to 24,000 being State of Mass. Rifles. There are also a few rifles in the 18,000 and 28,000 range, which are probably US Army contract replacements.
M-1865 Spencer Mfg. Carbine serial numbers – 1 through ~24,000
M-1865 Burnside Mfg. Carbine serial numbers – 1 through ~34,500