Who makes the best 1861 Springfield repro?
Depends what you mean by best? Are you looking for the most historically accurate for say, reenacting? Or are you looking for the best quality shooter?
As stated above ArmiSport and Euroarms make decent versions and both are widely used in the reenacting community but both have small issues with historical accuracy. I prefer the ArmiSports over the Euroarms though. There are quite a few people out there that "defarb" both rifles and do make them much more historically accurate by removing modern markings, refinishing the stocks, blueing the springs, screws, sites, etc. I can provide you with a few good ones if that is what you are looking for.
IMHO the best of the three mentioned is definitely the Pedersoli. I have one and love it. It's my shooter. I use a defarbed Model 1855 Harpers Ferry Rifle (ArmiSport), done by John Zimmerman, and a Model 1854 Austrian Lorenz by Loyalist Arms for reenacting.
The Pedersoli 1861 has a much nicer and fancier finish than the out of the box ArmiSport or Euroarms which isn't historically correct. Bayonets also need a lot of work to get to fit on the Pedersoli '61 but I did work one of mine with a dremel and eventually got it to work (just for my own satisfaction, as I said I don't use this one for reenacting). The real nice thing about the Pedersoli and what puts it way above in quality terms is that the lock parts inside are case hardened and not the cheaper softer parts used in the other two repros. Very much worth the cost increase if you ask me and why my Pedersoli is the one for live fire. Shoots great too! I cast and lube my own .58 cal minies and have found the combination she loves!
Just for fun, here is me with my M 1855 Harpers Ferry Rifle taking a pic for a spectator.
Here is my Pedersoli M 1861 next to my Hawken. You can see the lovely satin finish (but historically incorrect!!)
Here is is with they bayonet.