So i agree with you that it is ODD that you did not get a MAJOR spark when initially connecting the batteries, if you did not use any kind of precharge technique.
It's also odd that flipping the inverter switch from off to on results in ANY immediate change at the battery. The power the inverter uses to run itself is tiny, <3a from those batteries. It would take several seconds for the inverter to actually become operational and present any significant load to the battery circuit. Maybe 10 whole seconds! And that's only IF there were a load on the AC output!
So if you are getting some kind of problem immediately upon flipping that switch, AND you got no spark on initial battery hookup in spite of having no precharge circuit, i'm beginning to suspect you have a defective bms. Because you got no initial spark on battery hookup, the inverter would have to have ALL of its capacitors failed for that to be an inverter problem. If there was an actual open circuit between the inverter's battery terminals and the capacitors, you would also get no action/change from flipping the on/off switch.
But, because the batteries are in series, ALL the current which would be creating that spark/inrush at initial hookup, would have to flow through BOTH batteries. If one battery presents an extremely high resistance, it would prevent that current flow from reaching an obvious 'big spark' level.
It may or may not be BMS related. The fact that you ever saw a "<3v" number is odd because when the BMS disconnects, you would see basically NO voltage on the battery terminals. When it reconnects, it would not "slowly recover to 12.5v". It would jump almost instantly to the full resting ~13.2v. So i suspect you have a loose internal connection either between the cells and the BMS, or between the BMS and the case terminals. If you could communicate with the bms over bluetooth you could probably figure out which one it is from that, but either way if it's a sealed battery it's a warranty issue and not something you would DIY repair unless you had no warranty coverage at all.