Hmm, not sure how that system works and if it has a shunt and can read the exact charge level.View attachment 258131
Not sure how that screen gets its info
Regardless, the voltage of your battery is why you're getting the issues, it's dropping below 11v when you're running the inverter.
That couple of amps of solar should still be raising the voltage too.
Could be a few things:
Soc is wrong and you're actually at 10% or 20% charge
Batteries aren't balanced and one is low charge, it should be pulling from them both at the same time (IE one battery at 10.5v and one at 13v)