For starters, if it's important to you, I think you should go. You're your own person, and it's important for you to respect your own needs and wishes. By the same token, try to keep her wishes in mind as well; even if they don't make sense to you, she has her reasons for not wanting to go, and it's important to recognize and respect that too. That baseline mutual respect and frankness is muy importante to the health of any kind of relationship.
Here's an idea: lacking a date, something my friends did was get together as a big group and go together, and just have a good time. I think going as a group is pretty plainly platonic, and she probably wouldn't feel as upset as if you were going with a single friend. Still, I'd discuss it with her first. Particularly since this Prom disagreement seems to be a stand-in for a bigger discussion about the relationship?
If you're upset that you never do things together, or that she's constantly blowing hot and cold, turning down dates and things, etc. that's a perfectly valid problem; don't ignore it! And again she's got her reasons, even if they don't make sense or agree with your own feelings. If you're both serious and willing to give each other the time and energy to talk about it and compromise, I hope you can find a good, mutually respectful solution, whether that's compromising on Prom or resolving to go your separate ways, or whatever. Just sitting on this Prom thing, both you and she will probably end up feeling wronged about this. And once that happens, it'll color any resulting discussion, and probably make it tough for you to hear each other, or come to an understanding of one another.
TL;DR It's lame to make this Prom a proxy for working out your relationship issues, when you could actually just talk about it.