Do I risk damaging the FR-4 by heating from the back side?
It's certainly possible to heat it from the backside. It's going to require higher temps to the board to do this, practically speaking. Copper is a good thermal conductor, but that is only a small part of the board. The FR-4 is a good insulator, and the bottom will reach a higher temperature if you heat from the back. The typical way to heat from the bottom is with a hotplate to spread the heat evenly and to bring the board up to temp gradually and carefully. This would be rather tricky to do with a heat gun, IMO.
Hot air and paste is viable, yeah. You should want some extra flux however you do it, for post flow cleanup if necessary. Note, if you are using a paint stripping gun from the hardware store, the nozzle and potentially the air flow might be rather large. Heating a larger area of the board is not a deal breaker, but it's going to make it harder to get tweezers on the part without burning your hand. If the part skitters around due to flux boiling off, it may snap to the wrong pads. It won't stay still on the paste unless you ramp up very gradually and keep air flow low.
Pretinning the pads with solderwire and adding flux is another way to get the part on there. But you will need to manipulate the chip to keep it on the pads, even more so than with the solder paste. The smooth pretinned beads and boiling flux turn the part into a hovercraft. So you'll have to figure out if you can do this without burning your hand. What I do is just pin the chip to the pcb with tweezers using a proper hot air station.
https://youtu.be/mr1UVPsExiE?t=377In either case, you might need to go around the edges of the chip with a soldering iron, after. And you may need to reapply flux, if necessary. In the video, he uses BGA flux and a lot of it. So he can install that chip twice and then some without adding any more flux.