First I bring good news: it is possible to "win" in endless mode, because endless mode isn't endless. It has 5 minute time limit. I guess it could be done by A-rank Romeo (beat the game in 50 or fewer battles, each mission is 2 battles). This guide is about how to gain the S-rank one, by beating last mission in 40 battles or fewer.
I found the best party would be Spartan, EntropyAlien and ShadowKiller x2. This can clear the last mission in S-rank most of the time.
Starting up: At the beginning as you would figure out already, buy two puppets. Your character line up should be:
Pinocchio, Pinocchio, Variant, Corpse.
The corpse (undead) is really a good tank, you may keep him on the front line until you have two ShadowKillers.
Train as much as your number of battles allow (count the remaining missions, each one misson cost 2 battles and you will need to win the last battle in 40 or less)
Upgrade the second Pinocchio to Striker first. And then another Pinocchio to Striker, Corpse to Eater. Lastly Variant to Star spirit. Your party now:
Striker, Striker, Star spirit, Eater.
Upgrade the second striker first and turn it into a ShadowKiller.
Striker, Star spirit, ShadowKiller, Eater.
Upgrade the Star spirit to a Spartan.
Striker, Spartan, ShadowKiller, Eater.
Upgrade the last Striker to ShadowKiller.
Spartan, ShadowKiller, ShadowKiller, Eater.
Lastly, upgrade the Eater to EntropicAlien. Final party setup:
Upgrade the damage of the Spartan, then the ShadowKillers, finally upgrade the EntropicAlien.
Unless you are extremely unlucky, you should have no problem breaking through all missions in one go, getting the Romeo-S.
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Lastly, how to survive 5 minutes in endless? Even Romeo-S has 2000HP and 200 damage, he gets killed very easily. You need luck, because the ability Romeo has is random. Just retry until he have life gain for damage (a sword icon, same as the Spartan), there he will be invincible. Even if he had +60HP per second he may survive for very long (although he still gets killed eventually). You can see his current ability at bar in the top left during battle. Party setup: