Asking for documentation is a good idea if you can get away with it.
Her stories would seriously get to me. Can you (or have you) tried mentioning knowing this person or that one that might have been where she was? "Oh Dave was at XYZ during abc date. Do you know him?" Just to get it out there that you could easily check up on her if she gives you reason to. Maybe it will worry her enough that she'll stop bringing it up when you are around. She doesn't have to know that you already know. I assume you want to plausible deniability since she's seeing your friend.
First off Good on Ya for doing a Tough Mudder, I did my first one in the spring and another in September. So much fun
I think minus the asking for documentation this is the way to go. By dropping details about her deployments and locations, I think you'll find that she'll stop with all of the stories around you. Most likely a person like this is combining multiple stories and details into her own "narrative" that fits her life and personality. Ask questions and don't be afraid to mention if she is being inconsistent.
Peggy: CRUD MONKEYS! the sky fell down and there were evil attack butterflies everywhere
You: Yeah, my friend Dave was there and I thought he said they were frogs. Maybe he was in a different location than you
Peggy: JKL happened and then ABC happened and I was so scared
You: wait, I thought that ABC happened first, then MNO then JKL.
I had a "friend" that I had to drop a little while ago and they remind me very much of Peggy. They thrive on the drama and the attention. By asking questions and making it clear that I heard different versions of the "story", this person stopped talking about their "problems" around me. They would be very general, but they never got into detail around me again. Peggy is lucky in that she can talk the talk and kinda walk the walk. You'll never completely stop her from telling her stories, after all they get her a lot of attention. But you can can stop her from telling the stories around YOU