Had almost identical on my narrowboat last winter. BMV-712 showing 12.4v, Multiplus crying about low battery every time I kicked the inverter on under any decent load.
Turned out to be a combination of two things:
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Voltage drop under load — the BMV measures at the battery terminals, but if your cabling between the battery bank and the Multiplus isn't up to scratch, the inverter sees a much lower voltage than what the BMV is reporting. Worth checking your cable runs and connections, especially any crimps or lugs that might have a bit of corrosion on them.
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Low battery threshold set too conservatively in VEConfigure — mine was factory default and wasn't really calibrated for my Fogstar Drift lithium cells. Dropped the DC input low shutdown by about 0.3v and the false alarms stopped.
What batteries are you running? LiFePO4 and lead-acid behave very differently under load — a lead acid at 12.4v resting can sag to 11.8v pretty quickly if you're pulling 1500W+, and the Multiplus will absolutely throw a fit.
Also worth checking — is your BMV shunt wired correctly so everything goes through it? If any loads or charge sources bypass the shunt, your state of charge figures will drift and the readings become meaningless over time.
What does the voltage actually drop to on the BMV when the warning triggers? That's the key bit of info. Would help narrow it down.