Ended up getting a text to call them in regards to a claim, with a different reference number. I called the number, which was Sedgwick not Budget, the conversation went like this:
Sedgwick: Did anything happen to the car while you had it?
Me: No
Sedgwick: was there any damage when you picked up the car?
Me: Not that I noticed, nothing like what was shown in the photos I received
Sedgwick: Ok, I'll go ahead and close this

She didn't know why it was a different claim # but confirmed it was the same vehicle and assured me it was the last I would hear about it, and sent me an email stating as such. The funny thing was I never had a chance to tell anyone about the mileage discrepancy, so they somehow figured out it was a BS claim on their own. The conversation I wrote above is literally all we said to each other regarding the car.
Also I googled the phone number that texted me, trying to figure out if it was Budget or Sedgwick before I called it. Dozens of results came up with similar stories posted on this site, Reddit, etc. There was even a consumer watchdog article about someone who had this happen two years after they had rented the vehicle.