I use Woodland Scenics glue

I use Woodland Scenics glue and long t-pins to mount the tortoise directly to the foam.  The t-pins hold the tortoise until the glue dries, after which they can be removed.

I put a heavier wire on it than comes with the package because of the flex associated with depth of the foam.

I previously tried double-stick foam tape, but it failed on a few of the tortoises, so I resorted to only the glue.

When I used to use under

When I used to use under layout turnout machines, I would cut small squares of Luan, soundboard, or other 1/8" board and glue them to the foam under the layout with PL300.  The bigger square you can use, the better as it gives more surface for glue to help resist torsional stress.  I put a couple screws in to hold it until the glue dries.  You can cut the slots before you glue to make it easier and then just screw the machine up to it.  PL300 is the best stuff ever for working with foam.

I think you are right.

Scrap Masonite

I cut some 2"ish x 4"ish pieces of leftover hardboard, drilled a big hole for the wire, and then caulked them to the bottom of the foam (painted black by previous module owner in the photos below).
T-pins hold the hardboard in place until the caulk cures (about 1/2 day to make sure).

I've done it both ways:
--mount hardboard first, then attach Bullfrog OR
--mount Bullfrog to hardboard, then caulk to foam

Either way works, and perhaps mounting the Bullfrog first makes alignment a little easier.

If you have a few Bullfrogs close to each other you can put them all on one piece of hardboard (as I did on the left):

Notice that the hardboard doesn't have to be square, just long the width in the directions you throw the Bullfrog (shouldn't be applying force front or back, just side to side!)

Hope this helps.