I was told by my dealer not to overuse the FBH because I'll end up with a flat battery.

I have chosen to ignore this so far and I'm using it up to twice a day at the moment and, touch wood, so far haven't had any low battery warnings like I had on the L322.

Indeed, even the start stop system is still working which, in experience, is the first to stop working when a battery doesn't have a lot of charge.

So how much battery does the FBH actually use? Is it just to start it?

I believe fuel is only used to provide the heat. The rest is battery (startup, blowers and all the other electronics in the system).

On my L320 I occasionally used to get "battery low", "start the engine" and "system will shut down in x minutes" type warnings on the infotainment screen after using any of remote, timed or park heating. On a couple of occasions park heating refused to start due to low battery (I found the cause only by reading faults using my IIDTool). I always obliged and did what it asked so it never got too severe to stop the car from performing its main duties. It had a Start/Stop button but no Auto Stop/Start so don't know if that would have been affected.

I never got those warnings again in that car after I invested in a trickle charger and started leaving it connected when not in use. Been using that charger in my L494 from new so don't know if those warnings would have appeared otherwise. I typically use remote heating more than once a day on weekends. Current: Volvo XC90 T8 Inscription Pro PHEV with some optionals on top
Gone: MY16 RRS2 3.0 SDV6 Autobiography Dynamic, Carpathian Grey & Santorini Black Sliding Pano/Ebony & Ivory 2 Tone
Gone: MY13 RRS 3.0 SDV6 HSE Black Edition, Orkney Grey/Ivory
Gone: MY09 RRS 2.7 TDV6 HSE, Santorini Black/Almond