Thanks so much everyone for the advice, I picked up a few welding tarps and put them around the area. I used a die grinder with a carbide burr instead of and angle grinder with a grinding wheel to clean up the welds, made it much easier mess to clean up, as there is no abrasive dust and the die grinder makes larger shavings instead of metal dust. Still messy, but much better than with an angle grinder and abrasive disc.

I still needed to use some flap discs to blend everything together and that's not great, but the bulk of the metal removed is somewhat contained.