I don't think anyone here is saying his vacuum is drawing 12kW. The battery doesn't care for the load amount, overcurrent is measured in amps only - and if it exceeds 250a it will turn off in .1ms.
No, we're just saying it appears vacuum plus inverter drew > 250A from 48V battery. (however briefly)
250A x 48V = 12,000W
.1ms? as in 0.1ms, 100 microseconds?
Fast cutoff like that may be just what is needed to prevent MOSFETS from being killed by a short circuit. It is about the spec for a 100A MOSFET. So I'd rather have higher current or multiple MOSFET in parallel.
This is the basic problem with semiconductors for power circuits.
It is possible to have complete protection, but it requires inductors. An inductor in series with battery or other power source will deliver a current ramp rather than instantaneous short-circuit current. A switch-mode power supply can then operate as CV/CC, and never exceed maximum allowed current.
The "400V" batteries like LG RESU-H worked this way. 48V (?) battery and boost converter deliver 400V with limited current. That way they never exceed the 30A short-circuit current some inverters specify.
BMS doesn't implement such inductor. Inverter does or could. OP's inverter probably can deliver a high surge current, and draws a high current (maybe insufficiently filtered.) If an inverter could be configured to draw a programmed maximum surge from battery, we would have greater options to mix-n-match. Brute force sufficient battery is all we've got for now.
My 120V 6kW Sunny Island has spec to deliver 180A for 60 milliseconds. That is 21.6kW, 450A from 48V. At peak of sine wave I would expect sqrt(2) higher, 636A. I have four such inverters connected to a 405Ah battery.
There is a reason I like lead-acid
5C discharge rate for an instant? Oh, I didn't even notice. Lithium can handle that too (portable jump starters). BMS can't.
Do you realize 636A doesn't even exceed the CCA rating of a modest size automotive starting battery?