If you follow your plan and end up in a slip, you're no longer a "trailer sailor", so you're in the same position as the rest of us here in Michigan who slip their boats.
Oh...and by the way, "bottom" is more appropriate than "hull". Yeah...nit picker, but "hull" also includes the topsides, which plenty of people paint, too.
Anyway, the leading bottom coating by far in Michigan is Interlux VC17 or one of the equivalents from West, etc..
Pros: It's a snap to apply. It goes on very thin, so you don't get a lot of buildup. It's a fast bottom. Once applied, when you reapply it next Spring (or if you use it for two seasons), it requires very little prep. Plenty of info out there on the net about how to apply it.
Cons: It's a little pricey. But you're only going to use one can, and might even have a little left over for next time.
Yes, you would have to remove the paint that's on the bottom now. Some paints may be compatible for over-coating with VC17, but even so, you're going to want to get rid of 95 percent of the old coating to create a nice, smooth base for the VC17. It's not that bad a job. If I was doing your boat, I'd remove it with the a fiberglass compatible chemical strip. Messy, yeah. Then I'd sand it (wet). Probably with nothing more aggressive than 400 grit. Obviously, you want to remove paint without removing gelcoat. You could use 800 after that if you want. Then, on goes the VC17. Done.
Alternative: You could put on a barrier coat of your choosing before applying the VC17. Your call.
PS
I'm sailing a Catalina 28 currently, with a VC17 bottom. I find that it's so easy to prep and apply that I do it EVERY Spring. I carefully tape the waterline, scrub the bottom with a Scotch Pad, rinse, let dry or towel it if I'm in a hurry, and apply a THIN coat of VC17. I never get any buildup, and the zebra mussels and the algae leave me alone. And the bottom is smooth and beautiful!
Okay, one more thing. To do the job, ideally, you're going to want to raise your boat off the trailer. More than one way to do that, but if you're ever interested I can tell you how I used to raise my Precision 23 off it's trailer so I could get everywhere including the swing keel.