FWIW, I reckon regular riding is a good start - when I was commuting a reasonable distance (about 7 miles each way, 5 days a week) my role of thumb was that doing my weekly distance in one go was a fairly plausible ride. Maybe type 2 fun, but doable without *too* much trouble.
That was also my experience. My first Audax* was three times longer than my previous longest ride. I jumped from a couple of 30 mile CTC easy rides to the 150km Forest Ghump. But I'd been commuting 5 miles each way for over a year. I did struggle a bit, but not to the point where it looked like I might not finish, and the BP minimum time is generous enough to allow walking up any hills. Rode that and a few others up to 300km on a bike not dissimilar to the OP's, though the advice to change tyres was much appreciated.
I then spent a decade, and a considerable chunk of money, chasing the perfect Audax bike before ending up with a bike similar to the one I started out with, though of far better quality. I think the OP should ride more on the bike they have, with better tyres as the first upgrade, then look for a better bike for them rather than better for Audax.
* I didn't know what an Audax was, possibly still don't, though I now know what I want from it.