I agree with trying a small area to see what you have (keeping in mind that if it does go well, you still might find some "surprises" elsewhere on the trailer). Carefully scraping or wet-sanding would be good ways to start (Interlux Interstrip 99 is a marine stripper, but I have had it damage surfaces, so be careful if you use it).
With the "quality" of the paint that's already on there, you won't have any trouble covering up your tracks if you do decide to cover up your experiment and stick with your current paint.
If you're going to do any work with stripper, acetone, paint, etc. I'd just go ahead and buy a good respirator. They're not that expensive, they come in handy more often than you'd think, and lungs are priceless. Note that the decent ones from 3M, Survivair, etc. come in sizes, and small will nearly certainly be what fits you (it's barely small enough for me, and I don't have a tiny head or anything; just a female head).
As far as "never being able to go back to gelcoat," in my opinion, while it's great to have original, nice gelcoat; once it's gone it's nothing I would lament not being able to go back to. A good, two-part LPU paint job is in many ways better. I know I'd choose that over "re-gelcoating" any day.
The one-part paints, such as Interlux Brightsides, are good paints, and relatively easy for the DIY-er to apply. The caveat with them is that they are softer than two-part paints, and so somewhat more prone to chips and such, and, while they are nice and glossy at first, that gloss will not last nearly as long as that of a two-part paint. Mostly that's from UV exposure, so if you keep it in a garage 90% of the time, that property of one-part paint may not bother you. (Or if you don't mind re-painting every few years.) The paint is less expensive and easier to work with.
On the other hand, most of the work is in the prep, and a good two-part paint job will last a long
time. My boat was painted with Awlgrip two-part LPU in 1985 and it's still glossy 23 years later.
An additional note is that you cannot safely spray a two-part paint without a supplied-air respirator system (which probably means you should have a shop do it); although you can
roll and tip a two-part system with a regular respirator (the most dangerous is when the paint is atomized by the sprayer and makes a toxic mist in the air).
All that said, it sounds like anything will be an improvement, and I realize you're not probably going for the "gold plater." Who knows, maybe the PO just didn't like Robin's Egg blue and you'll reveal a nice gelcoat underneath