I don't like heaters inside enclosures much I purchased a RHP to play with and I have used a product similar to a Kane mat in testing. The mat I maxed at 96º I never got it hotter than this and Kane has a killer rep they make I think hands down the most durable heat mat available (and most expensive). From my experience with both the RHP is scarier. I am going to fiddle with it but...

I understand your problem I am leery of using any heater inside an enclosure but if you are going to I have a few suggestions to make it as safe as possible. I would NEVER use an on/off T-stat only a proportional unit The probe must be very firmly fixed to the pad. It must not move at all. I use hot melt glue but I would be concerned about it softening on a heat mat like this so I might investigate other glue systems (3m 5200 comes to mind...)

I would also always use a On/off T-stat as a fail safe in addition.

This is a fairly expensive operation at this point, I am going to make a second suggestion replace the bottom of the enclosure with PVC (not expanded or foamed just plain PVC) it transmits heat well and efficiently you can then use traditional UTH systems.