If a generator is a must have, buy a boat with one already, it is by far the cheapest route. That said, as a fellow Chesapeake Bay trawler owner, I will say that I run the generator less often and frequently for different reasons than I would have expected prior to getting into trawlers. I will run it underway when I have the whole family onboard and everyone but me is in the cabin on their electronic devices. The only way my single air conditioning unit will keep up while the engine and generator are both radiating heat up into the cabin, is with all of blinds/shades closed. Meanwhile, I'm completely comfortable up on the flybridge with the bimini for shade.
Anchored out, with the windows and hatches open the boat is comfortable on all but the hottest of days (93 degrees and above). I'll run the generator during dinner and cool down the cabin prior to going to sleep but shut it down before I fall asleep. In a marina without air conditioning, the boat is much hotter, there isn't anywhere near as much breeze and there is a heat island effect so I'm always plugged in and running air conditioning when in a marina. The way most trawlers are configured now, the are pretty significant 12V loads at all times between refrigeration, electronics and the heavy hitter is an inverter for 120V systems. I use my generator to recharge my house battery bank during breakfast and dinner when achored out more than I run it for air conditioning. If I was starting from scratch, I would just focus on solar, a larger house bank and heavy duty alternator setup. This would be a more cost effective approach rather than installing a generator, and for the days that are just too hot and miserable to be without AC, you can pay for a lot of transient dockage with the $20-$25K that a generator install would cost. Before the jellyfish coming into the bay, I'm more likely to anchor out and swimming off the boat can cool me off if there is no breeze, once the jellys roll in, I'm much more likely to take a transient dock. My small genset burns about 0.5-0.75 gallons per hour so I could easily burn $30-40 in diesel keeping the cabin cool on a hot day, that goes a long way in offsetting a transient slip.
In 2015, my parents bought an old 34' Mainship, their first "trawler", it didn't have a generator and it was one of the first priorities for us to install. We found a bargain on a new, off-brand unit itself, a 4.5kw and did all the install ourselves, it was about $10K all in which was incredibly reasonable. I know we couldn't do the same today for less than $15K. Later that season, we installed an autopilot because the boat didn't have one. The AP was a game changer and 1/5 the cost. We would have never guessed the end values of those two items prior to running a trawler and the whole family has a lot of boating experience. Now I have my own trawler, basically the same as my parents' old boat, it came with a generator and I use it but I wouldn't spend $15k to have one.