I always determine the piece values by play-testing from initial positions with unbalanced material. Just give one player two Captains instead of Knights in the FIDE setup, and let the computer play a couple of hundred games, and look which player scores better (and by how much). From tests like that I obtained
second Bishop > Lieutenant > first Bishop ~ Knight > Captain.
Although the differences are not very large. Warlord plus Pawn is slightly better than Queen, and the spare King is better than any minor, but worse than a Rook. The General is about halfway between Rook and Queen. Persian and Hoplit Pawns are about equal in value (but depends a lot on the Pawn constellation).
I must have published the EGT results on some forum (probably TalkChess); some are here: https://www.chessvariants.com/index/listcomments.php?order=DESC&itemid=MSspartan-chess . Redoing them would probably be faster than trying to look it up; I use FairyGen to calculate the 5-men end-games. (It can only do 8x8 boards, but that happens to be OK for Spartan Chess.) For 3-vs-1 (or 2-vs-1) EGT yu can try the Checkmating Applets at chessvariants.com.
"The Captain has mating potential, despite the fact it isn't worth more than a minor,.... End-Game Tables have been constructed for all pawnless 5-men end-games.
How do you determine the values of the pieces? I always have difficulty deciding whether to exchange the FIDE B/N with the Lieutenant/Captain when playing the computer.
And can you point me to the End-Game Tables? Try googling it without success.