Cushioning... controlling movement is most of the battle here. You do not really need cushioning as much as you think.
Yes epoxy is the right choice for use with foam... polyester resin will melt the foam and generally NOT work the way you intended. I have successfully used polyester resin with foam, but I mummified the foam with duct tape first. I have heard of folks painting the foam with latex paint to seal it as well. Those techniques are most applicable to removing the foam core.
The removal aspect... the wheels was what I was thinking when I mentioned only removing what you have to. With a sanding drum on a drill you should be able to produce some nice arches matched to the wheel circumference and still keep the foam sandwich construction on most of the box. The foam sandwich is typically what is used in aircraft composite structures due to its combination of light weight and strength. By moving the structural elements further apart you get much better strength, glass or carbon fibers are best in tension/compression and poor in bending. A single layer of fibers puts most of the stress in bending where the fibers are weak. In bike terms think little tubes versus big tubes: big tubes light and strong
BTW this is also why I thought stiffening ribs would be beneficial.
There are a couple directions you could try going on the seam between box and lid. One is to do the layup as one piece and then cut them apart. A fine tooth jigsaw could do this but even better would be one of the vibrating saws that have been the rage the last couple years. Those have suitably fine teeth and do plunge cuts. Woodworkers have been known to run a cube through a table saw to create a box/lid. The other direction is layup one side and then use as part of the form for the other layup. Saran wrap between layups will keep them from gluing together. Same for duct tape with the advantage of a bit of thickness for future clearance. Using this technique you could create a bit of an overlapped seam.
And speaking of saran wrap... on a flat panel layup like this plastic like saran wrap is fabulous to put over the top of the layup and work the resin around under. Can get almost vacuum bag quality that way. You might find some very interesting videos on the subject at the Expiremental Aircraft Association (EAA) web site. They have one using a baggie to create a pre-preg of sorts for hand layup.
And one last thought... do not mix resin types. I made this mistake years ago putting polyester over an epoxy joint and had a devil of a time getting the polyester resin to kick off over the epoxy. So once you start with epoxy you will have to stick with epoxy. You *may* be able to put epoxy over cured polyester but definitely not the other way around.