I'm updating the thread for posterity.
Blackstone's support team was very helpful in resolving the issue. They responded within a day of my emailing them and sent me a new base (the main part) of the stove, as well as a new heat shield. Some useful info from their mail:
I believe there was a leak in the valve. We had an issue with this a few months ago, but we thought we had isolated the problem. We have improved our valves in our ovens to prevent this from happening. We have also moved the motor down a few inches to help the motor last longer.
To lower the motor, they've added another spacer mount (the thing that the motor slides on to) on top of the old one, and included a longer drive shaft.
Another difference between the heat shield I had, and the replacement, is that the new heat shield does not have any paint/coating on the inside, it is just treated metal. So, no paint burning and flaking off.
My old valve was 10 PSI, the new one is 5 PSI. This doesn't seem to affect the stove's ability to reach extremely high heat, just seems to reduce it turning into a massive flamethrower shooting out the front from the moment you turn it on.
Assuming I'm not posting back here in two weeks saying my stove caught on fire again, I'm pretty happy with how they've handled this and still recommend.