Mike,
I am not a reenactor, but I would first say to research the unit you want to portray. Were they armed with Enfields? Also, I shoot my original L.A. Co a good bit. Not everyone can get comfortable with the way an Enfield shoulders. I would suggest handling both the Enfield and Springfield and see which you are more comfortable with.
If your unit prefers an Enfield, a L.A. Co would be the cheapest to replicate, as the lower band is small like the reproductions, the washers opposite the lock are round eared.
If my memory is right, in emails with Mr. Walden, I believe L.A. Co. started using the 4th pattern band in mid/late 1861 (Mine is 1860 dated)

As to if it matters, IMHO that really that is up to you. Most of the general population haven't a clue the difference between one properly marked, and one not.

Virgil