If you do some searches here, you'll find a few tales of other people dealing with similar problems, as you suspect.
Before you repair the floor, make certain you've figured out where the water was getting in. Typically that's someplace where something penetrates the roof (a vent, skylight, etc.), or is attached to a wall (clearance light, window, etc.), or a seam where the roof and walls meet. That also means that often there's damage to the wall and/or ceiling structures as well as the floor. Take a good thorough survey of it all before embarking on a repair project so you know what all you're in for.
Also be aware that often the floor covering is applied before the cabinetry etc., and extends underneath them. You may well have to remove a good bit of the interior pieces of your trailer to repair things. The good news is that often they are more or less just screwed into position, so this isn't an impossible task by any means.
I would resist the temptation to reengineer the floor much if any. Thicker plywood would mean a heavier floor, and cut down on your capacity for carrying water or cargo. (I don't know how it is on your trailer, but I would guess that you don't have an excess of weight carrying capacity to begin with.) The laminated plywood and foam sandwich is quite stiff and strong when properly made and bonded together; it acts a little bit like a truss, with the plywood carrying the loads mainly in tension and compression, held in position by the foam. I'm not sure that aluminum joists through it would actually strengthen it much, and if they prevented bonding with the styrofoam they could potentially weaken it somewhat. (They'd also reduce the insulation value of the floor, but that's presumably not a big concern for a hybrid trailer.)
For making the plywood and foam sandwich, I'd probably use epoxy, possibly with a thickening agent for better gap filling, and lots of distributed pressure, applied from the middle and then working outwards. As I'm certainly no expert here, that may not be the best approach or adhesive to use. I definitely would suggest at least looking into using marine plywood; it's far more resistant to coming apart when wet than exterior or especially interior plywood, and usually is made from better quality veneers than ordinary plywood. I would keep the bottom waterproofed, though if there's a convenient way to do so it would not be unreasonable to leave a little airspace between the bottom plywood and the waterproof covering. The bottom will get quite wet anytime you drive through the rain, and is exposed to rocks and sand and debris that wear away wood finishes.
Pay attention to how the walls and floor attach. I think often the wall structure is built on top of the floor, and so any part of the floor that needs replacing underneath the wall will require some careful thought and work. There are many variations in such construction details, though.
This sort of a project is a rather long and difficult and involved one, but it's certainly not beyond the reach of a fairly handy and dedicated do-it-yourselfer. It may end up being somewhat costly; take a sober look at things and decide if it makes sense to repair the trailer, or to get out before throwing good money after bad.