The thing's you've planned so far are all great for dates. For bars...Sub 13 is a cocktail bar at the top of town, if you're into that kinda loud music cocktail bar it's pretty nice (2 for 1 cocktails until 8 most days too!). Great heated beer garden too.
As for restaurants, if you really have no budget the Royal Crescent hotel has a pretty upmarket restaurant, plus it's in the Royal Crescent so amazing location. And if you're looking for a coffee shop/cafe to take a break during the day, I can highly recommend Jika Jika. By far the best coffee in Bath, plus I hear the food is pretty great too.
Enjoy!