Simple solutions are sometimes the best - instead of wearing mitts, try covering the bars with a towel which you wash with kit after every ride. It deals with that nasty sweaty / sloppy mitts feeling and protects the bike from the sweat.
i’ve been a fan of the endura fingerless gloves for years. they make a pair that has perforated pads on the palm for cooling which seems to work pretty well. They’re at least more breathable than gloves that don’t have this feature. Pearl izumi also makes some gloves like this. I use an old pair of the enduras that have velcro that no longer works on the trainer with the closure strap cut off. Never felt like they accumulated too much moisture and have that little bit of extra padding that’s desirable on longer interval workouts.
I raise front wheel a bit to remove some pressure from hands (less padding needed) and then i cover bars with whatever material i like, microfibre towels can be pretty fine, you can use them also to wipe hands during a ride.
I keep the same bike on my trainer the whole time so my solution is easier than in a lot of cases. I have wrapped a load of foam padding and duct tape around the bars. It looks hideous but is much better for comfort now.
Honestly you might need a better fan. I wear specialized body geometry full fingered gloves and with my lasko no problems on 3 hour rides. I like full fingered as they don’t ride up on my fingers.
I have a stack of about 12 hand towels for riding for a week. I generally keep 2 handy during a ride. 1 I drape over the bars and hoods and one as a backup on a stand nearby. On bigger/harder days, I’ll keep 3 nearby and after an hour I’ll throw the damp one on the bars on the floor and replace it with a fresh one. No complaints about this setup. I should also say, I have a sweat net on the bike also so the towel kind of rests on the sweat net as well, which keeps it off the top tube and relatively flat.
If you can find the previous generation the section just below the fingers used to have the same gel stuff as is on the lower sections. Similar model number and aerogel designation. Changed it for some reason, who knows but I’ll probably give them a shot anyway when my current pair wears out.