I'd agree that fisher can be the biggest money maker at the moment, and possibly with the right supporting class.
Big fish are bringing big money.
Culinarians who want to level desynth need your 2 star fish. I'm in the low 80's for CUL desynth and get about 0.2 of a skill up from a 2 star fish. If I couldn't fish my own up the process would be costing me millions of gil.
It isn't just people levelling the desynth that want your fish either. The ones who already have it levelled are buying them to desynth, hoping for the rare and extremely valuable drops. I've seen some of them listed and bought for millions. If you have cul desynth levelled you could go down the route of gambling on desynthing them yourself or you could take the safer route and just AH list your fish.
Relatively easy to catch fish like gigantsharks sell all day long for about 75k on my server. Something like a Frilled shark would go for much, much more. There are still fish we don't know the drops from and which haven't appeared on the AH listings. You could probably name your price for them.

The downside is that with FSH you're playing the rng game big style. Botany or miner vs fisher is like mnk vs blm. Botany and miner will give you steady income, fsh will give you lean patches and big big spikes.