“If I have $50 of Bitcoin on the Lightning network. I don't care about custody.” yes，custody Lightning network is shitcoin.

I disagree. Happily used it every day in El Salvador. And it works. And it's bitcoin. The only use case I can think of for e-cash is CBDC and dog meme coins.

I would imagine the relay operates the mint and since lightning is used to get in/out your client could just use lightning to buy the exact amount of ecash needed for a stamp and then use that? Probably still prohibitively expensive with lightning fees for one stamp but maybe you could just round up to whatever makes sense whenever your balance is too low.

Sure, but the mint can still issue more tokens than it has in reserve.