I like the idea of coins as time. Reminds me a bit of how arcade racing games give you a certain number of additional seconds every time you hit a checkpoint. The king should reward you with enough coins to survive the next level, even if you're close to zero; the bonus you get for finishing with extra time/coins is that much extra survivability in the future. This would put an interesting spin on any opportunities to spend money on equipment, if you wanted to introduce that (don't know what equipment would mean in a 1HP game, but these ideas could apply to other games) -- invest in power so you can survive the enemies, but spend too much and you may lose on time.
Of course, the down side to carrying over your success is that you also carry over your failures, eating up your hard-earned surplus all in one go. I'd say, though, that's no different from, say, burning through scarce healing potions too early on in BRogue.