Maybe I'm just an old man now, but I feel like you're overthinking it. If you feel it's worth another try call him and take the initiative to suggest something. If not, don't.
Don't ask strangers what to do when you know what you want inside.
My first date with my wife was us with a group of mutual friends at the zoo, then she pretty much decided I belonged to her. So I'm not really one to tell you who should pay or whatever.