Not to disagree with Dave, but on a structural note I would say that in this limited area where the ply is there to make a lower standing surface the act of bonding the bottom and the top to the ply and the sides would stiffen the structure.
If you think of the surfaces as being in tension or compression ans the wood just keeps them a fixed difference apart (along with its stiffness since this is what the plywood is made for with the crossing plies.) I would think that the bonding would be a good idea.
My question would be how was what you removed installed?
If it was bonded then I would redo it the same way.
It it were not bonded then the same thing might well be the answer.
Since the floor is already in place I would leave it and add two 1" aluminum angles underneath and across the floor to stiffen it.
I would also bond the floor in place with
fiberglass cloth covering the floor.
I would not personally use mat, but rather cloth since it will lay down and make a smooth surface and since I have Epoxy I would use that, thinned slightly to soak a little into the wood for the coat over the wood.
I would add the cloth (precut and fitted) and use unthinned Epoxy to wet out the cloth.
It will leave a clear
fiberglass coat over the top, well bonded to the wood.
I would the lay up some strips to bond the floor to the side kicks.
If you are careful and lay the strips carefully it will require little finishing and the floor itself should be smooth enough to put the flooring over directly.
However the addition of some steel or aluminum angle bolted through the floor with flat head bolts will stiffen up the floor. I would bond the angle to the bottom with a little polyurethane glue/sealant.
The secret is bonding to prevent movement of one part relative to the others.
There is strength and there is stiffness and they are not the same thing.
All of that being said I probably overkill most things I work on so my trailer ended up a little heavier than many others, but I don't worry about it falling apart, either.
After rereading and looking at your pictures I would say that one angle across the middle under the floor would take care of your bounce. The also a piece of wood, bonded and fiberglassed underneath would also do the job. This could be tapered at the ends if you like.
I think a painted steel angle, bonded and bolted would be the trick. 4 bolts one on each end and two spaced in the middle.
Be sure to seal the holes and bond the angle to the
fiberglass to prevent movement.