10V? That IMO is too low. Usually I see 10.5V and even that can be a bit low, especially if it's a relatively low discharge current.
Even with UPS systems that I used to deal with (eg, I'd devise specs and evaluate tenders), though we could go as low as 9.6V we usually opted for 10.5V, and that assumed the batteries could be recharged ASAP. And of course, no engine cranking was involved! And that was with a 10 or 15 minute discharge (from fully charged to its endpoint of eg 10.5V); it's endpoint voltage would be higher for slower discharge rates.
You may be better off with a PC etc type PSU like the Carnetix, DSATX etc as found on this site. (Maybe not the M2 & M4 series etc due to their reported noise problems.)
They have programmable shutdown voltages as well as ride-thru features (ie, to survive cranking voltage dips etc).
Otherwise maybe a convertor from places like current-logic.com, and if they don't have a suitable shutdown, maybe a battery protector like the ~$20 MW728.
(Funny, I just replied in another thread about the MW728! Elsewhere I have explained how the MW728 can be tricked into higher cut-off voltages than its standard 11.5V and a way to negate that proportional increase to its turn on voltage of 12.5V else have a different increase for that. And as I recall, those mods each require an extra relay plus a diode or few, and can also prevent an on & off cycle of at least 15 second cycle periods that can occur, though a single relay can also prevent the same.)
The above is irrespective or in addition to any isolated two battery setup. (A UIBI system I hope!)