Epoxy to the rescue...
I would coat the inside surfaces of the hole with epoxy. Even perhaps dig out a bit of the styrofoam and make a tube out of blue tape and put it in the hole so it makes a dam of sorts.
Then get a glue syringe and fill the void with epoxy. You would be creating a plug of sorts where there is enough solid area to cinch a bolt against.
A derivative of this technique is common in the boating industry.
Like this:
Might be a fairly involved project though - especially if you remove the doors to route out the voids.