IANAL, but my two cents:
I think the restaurant would be totally right in demanding the full bill be paid by whoever’s at the table, because it’s unfair to make the restaurant figure out how the diners split the check. When I go out with my friend, the restaurant doesn’t care if I pay for dinner, or he does, or we split it-it just wants the bill paid.
More importantly, it’s not the restaurant’s problem if we disagree about how to split the bill. Again, it just wants its bill paid-and it’s unfair to make it enforce an agreement it didn’t know about and doesn’t care about.
How is the restaurant to know what the agreement was? It wasn’t a part to the agreement-so I don’t see why it should have to enforce it.
Also, as you note, it looks exactly the same to the restaurant’s point of view if A was expected to pay it all, B was expected to pay it all, or they agreed to split—two people ate, one is here, and she isn’t willing to pay the whole bill. It might reasonably suspect that the OP and his date agreed that the date would pick up the check, the OP left, and that the date is trying to get out of paying the whole bill by blaming the OP.
To change the OP’s example in one little way (to make this clearer):
A and B go out to lunch, and previously agreed to split the bill 50-50. A gets caviar, steak, and champagne, while B has a green salad. Say the total bill is $200, and B’s salad was $10.
The bill comes, and A puts down $100. B puts down $10.
A says “I paid my share, and won’t pay more-that was the deal”
B says “I paid for what I ate, and won’t pay more”
All this does is change the OP by having B pay $10, rather than running out the door. A still didn’t expect B to sneak out on the bill.
This example seems to make the answer clearer-why should the waiter have to resolve this fight? He just puts down one bill, and wants them to pay it.
I think that if the date wants to get the OP to pay his share, that’s between the two of them-not between them and the restaurant.