You would think so. That's the configuration I had (I'm using a different inverter though) and mine went up in smoke. Admittedly when I first started using it, I had all of my DC loads connected through the BP including the DC side of my inverter and then I started reading about all of the returned units, and the mention on Will's channel that you can't have an inverter connected to it at all, so, I switched it to control only DC loads that did not include the Inverter (fridge, water pumps, etc) and a month or so after that is when it failed on me.
My suspicion is that the BP either doesn't have over-current protection, or, it does, but they didn't consider short / transient spikes in current which instead of just triggering OCP like you would expect, it just damages it over time until eventually it catches on fire.