Upon further research, a short on EITHER hot leg of a 2P breaker would trip the entire breaker (both sides). However, it would still take 50A to do that, which would NOT be safe to run through all the 120V wiring (which would only be rated to about 20A), including the 120V outlet, the tool's power cord, and the tool itself.
So yes - any 120V circuits on the cart should absolutely have some sort of protection at a much lower current (20A). I don't think a circuit breaker on a power strip would be enough, unless perhaps the power strip was hard wired instead of plugged into the 120V outlet. It would probably be easiest to just run 2 cords from the cart to the wall - one for 240V and one for 120V. I got a little ahead of myself - thanks for bringing me back into line!