To me, Jay comes off as a self-absorbed manipulative little piece of carp. Which is not as judgmental as it sounds, as most of us go through that phase at one time or anther. However, he's sixteen years old, he should be able to negotiate asking a girl on a date without having his Mommy try to MAKE her go. Cuz that's so attractive to girls, right?
DD was perfectly clear the first time she said I am not going to prom. Whatever he thought, interpreted, imagined, etc is not DD's problem. For crying out loud, insert something else besides prom in there, and there is obviously no consent. I mean, on paper "I will let you know" or "as I told you before, I am not going to prom with you or anyone else" would be clearer, but that's not the way people really talk. She did not mislead him. For him to lay out money on the basis of a conversation that didn't really happen, is at best irrational wishful thinking, and at worst extremely manipulative. It also has no bearing on how adult social etiquette really works.
He said, "let me know if you can go". He knew, when he left that conversation, that she did not intend to go. And he never mentioned it to her again. How is this appropriate? Would you cook an elaborate dinner or buy concert tickets for people who said they couldn't come? It's jut not reasonable behavior.
If Jay's mom has the nerve to call again, you can just tell her that DD said no, and Jay's buying ticket's etc is on him. Maybe she should be talking to her son about actually listening to what women tell him. (I would not say that part to her, of course). DD has nothing to apologize for, and personally I think that encouraging her to apologize for his imaginary conversation, is going to set her a very bad precedent for dealing with men in the adult world, both in business and in dating. DD needs to have the confidence that yes, she was there, yes she had half of that conversation, and she did not agree to go. You don't want people like Jay making her question what she already knows to be true.
Now, if Jay is going to tell his mom a made-up version of what happened, it's very possible he will tell others also. And yes, there may be people who will think DD was in the wrong, or is being jerky, or whatever. But that's going to happen at some point, in high school. People spread rumors. Innocent people get talked about behind their back. It's just something you have to learn to address. The simplest way is to say, "that is not what happened."
I pity the woman Jay finally decides to propose to if Mommy doesn't back off.