I do the same as you. Boiler is turned of once I add the bag and grain@Hazelwood Brewery do you leave the heat on when mashing?
I always turn it off completely and just wrap it up in an old fleece jacket. Usually only drops 2c over a 60 - 90 minutes mash, which I can't see objectively altering the beer.
My worry about using a false bottom and maintaining some heat source, is the mash will get hot at the bottomand without a recirc pump then it won't be an even mash temp.