I'm with Gary on this one, it seems like those batteries are a bit weak. For comparison, we have a set of the absolute cheapest LiFePO4's you can buy right now and they stay above 12.6v at .5c (equivalent of 200a with your setup). With a two foot round trip your 2awg wires should be dropping less than .2v or so and a bad crimp would have shown up as a warm lug and wire which you don't seem to have. You should be above 12v at the inverter terminals all the way down to below 20% on your batteries.
If it's not too hard to do I would check a few things to rule out the batteries or maybe find the problem. I would charge the batteries one at a time at 14.6v until the current drops below an amp and leave them there for a few hours to help balance the cells. If you never did this "out of the box" there are probably low cells in each battery. The BMS balances the cells in your batteries by "holding back" the higher cells with a resistor and bleeding off power while the others catch up. It only does this when the battery is above 14 volts. You can do this by setting the float on your charger to 14.6v. If the amperage going in the battery drops suddenly and the voltage spikes above 14.6 it means they were in fact out of balance and the BMS turned off. Once the battery has been at 14.6 for several hours so that you know its 100% charged and balanced run the inverter with an 800-900w load to cause a 100a draw in the battery and see what the terminal voltage is. Running at the BMS rating (100a) and .5c the voltage at the battery terminals should be above 12.5v until the battery is almost completely discharged. Repeat with the other battery and make sure they are both healthy, sometime when you have LiFePO4's in parallel one can trip it's BMS and not be contributing and it's hard to tell. With your setup if one battery trips, the other is not going to be able to maintain the load by itself.
Multiply these values by 4 for your 12v battery. You can see that the voltage at the cells should be 12.8v for almost all of the discharge curve. The BMS and skinny little wires they use where you can't see will drop that to around 12.5v at the terminals.
If it's not too hard to do I would check a few things to rule out the batteries or maybe find the problem. I would charge the batteries one at a time at 14.6v until the current drops below an amp and leave them there for a few hours to help balance the cells. If you never did this "out of the box" there are probably low cells in each battery. The BMS balances the cells in your batteries by "holding back" the higher cells with a resistor and bleeding off power while the others catch up. It only does this when the battery is above 14 volts. You can do this by setting the float on your charger to 14.6v. If the amperage going in the battery drops suddenly and the voltage spikes above 14.6 it means they were in fact out of balance and the BMS turned off. Once the battery has been at 14.6 for several hours so that you know its 100% charged and balanced run the inverter with an 800-900w load to cause a 100a draw in the battery and see what the terminal voltage is. Running at the BMS rating (100a) and .5c the voltage at the battery terminals should be above 12.5v until the battery is almost completely discharged. Repeat with the other battery and make sure they are both healthy, sometime when you have LiFePO4's in parallel one can trip it's BMS and not be contributing and it's hard to tell. With your setup if one battery trips, the other is not going to be able to maintain the load by itself.
Multiply these values by 4 for your 12v battery. You can see that the voltage at the cells should be 12.8v for almost all of the discharge curve. The BMS and skinny little wires they use where you can't see will drop that to around 12.5v at the terminals.