Would this homebrew battery storage solution work out?

I’ve been looking at some small AC coupled 5kWh battery solutions and the cost is prohibitive given that, from what I see, they need pairing with particular inverters so I’d be buying a new system: charger/battery/inverter to use with my existing PV modules.

On the other hand, a pair of 24V 100Ah LiFePO4’s can be had for around £1600 and I already have an unused 3.6kW MPPT inverter input. The batteries each come with a 24V 15A mains charger so would fill the batteries in around 7 hours. They’re isolated so they would float onto the two batteries, and the batteries would each connect to a boost converter ([email protected] in for [email protected] out) with the output of those wired in series to meet the minimum starting voltage of the inverter.

So, around 1200W for 3 Hours or so, to extend the self generated day a bit. All in price ~£1640

The only issue I can see is when the built-in BMS cuts out for undervoltage protection, one would trip before the other. Where would this leave the other battery?