Can we take a step back...
Don't you have trouble stirring the mash inside the bag? How do you get around the bag twisting around your mash paddle the whole time?
We know, stirring and turning over the mash is essential, because hot mash from the bottom needs to be brought upward to heat the cooler mash above it.
I'm curious, how well does that work with the bag being in there?
Due to the bag, you can't scrape the bottom of the kettle to prevent stickage and scorching/burning.
Hence I understand your idea of using a (thin) silicone mat to provide enough insulation to prevent the bag from scorching/burning/melting, while there should be enough heat convection through the mat to heat the bottom of the mash (in the bag).
On the other hand, using a false bottom would require some form of wort recirculation, to bring the hotter wort from underneath the false bottom to the top.
I have those stainless false bottoms from Brew Hardware. I bought the shortest legs, to minimize wort below the bag. Just enough spacing to keep the bag off the bottom. BTW, you can use them without legs at all. The false bottoms are constructed with rims that keep the mesh about 1/2" up.
As for stirring, I stir like I'm drawing a big asterisk on the bottom. Keeps the bag from twisting. And I use a big honkin' French whip, so it doesn't create as much rotation as a mash paddle would. Breaks up the dough balls nicely.