I started riding audax fixed because I only owned one road bike at the time, an On-One Pompino which I'd bought for commuting and then gradually built up to doing longer leisure rides. At some point around 10 years ago I heard about audax and thought I would try it, so naturally I just used the bike I owned. I really didn't know any better, and it worked fine! I would absolutely recommend it.
Those first few rides were a learning experience, and I had some interesting conversations with other fixed riders who I found out were always happy to talk and hand down the wisdom. Fixed audax is a niche within a cycle sport that is already considered tough, so there is a rapport and camaraderie within fixed randonneurs. Conversely, geared riders often assume fixed is harder than it really is (don't tell them)
One point to bear in mind about fixed audax is no matter how long, hilly or arduous a ride may appear, someone will have done that ride - or similar - fixed wheel in the past, so it absolutely can be done. If you're the type that likes to have a goal, check out the AUK Fixed Wheel Challenge to work toward an award:
https://audax.uk/awards-pages/fixed-wheel/As to the choice of bike, I would say if you are comfortable riding your FXE for some hours at a time at a steady pace and comfortable with your choice of gearing up and down the hills you'll face, then that setup can absolutely work fine for audax. Being comfortable at longer distances is IME a case of trying it and then sorting out any niggles or pains.
Gearing is a very personal choice and I tend toward low to medium gears, even though I ride mostly in the flat terrain of Essex, Suffolk and Cambridgeshire. Here are some examples that work for me:
Flat stuff - 73"
'Do everything' gear ie most of my rides - 69"
Hilly rides, or I feel like taking it easy - 66"
Relentless sharp hills eg. Sussex / Kent AAA rides - 63"