I'd think the false bottom would take care of that. Stirring should help. I do partial mash/partial boil biab. I put a cake cooling rack in the bottom of my BK to keep the bag off the botom where it'd burn. So stir it occasionally,that should help.
You may get some grain beneath the false bottom and it can scorch. Also, the false bottom will impede natural convection. You can't stir the water that's under the false bottom. It would be better if you'd recirculate. Recirculating will clear the grain from beneath the false bottom. You'd be pulling the heated water from beneath the false bottom and running it through the mash. Your 1 tier system will be more complete with a pump, gas valve and a temp controller.
Yes, if you have a pump, just recirculate at as fast a rate as you can without sticking the mash. With a false bottom, once the mash has sat idle for about 10 minutes, I turn on the pump and slowly open the flow (March 815) to about 1/4 open on the control valve. I run my liquor to grist ratio pretty high (about 2+ :1) and have no problems with scorching or stuck mash (even with wheat malt and no rice hulls).