Alternately you could take the bms out of the inverter/charger current path.
Use the bms to remote control the inverter/charger.
I'm not sure if that inverter/charger supports it but it makes a sweet solution.
1500W @ 12V is 125A. Something higher to cover inefficiency and running off a low battery, but 200A should be sufficient for fuse.
Size fuse for continuous current. Size wire at least as large as the fuse (2/0 awg in this case), maybe larger for surge current and voltage drop. If wire is very short (1 or 2 feet) should need to be oversize.
Size BMS for surge current (375A)
If not, how about just mounting an outlet with neutral wired to ground, nothing else? Plug the power cord into this outlet when not on shore power.
(I suppose we could fine some degenerate case where somebody gets electrocuted the moment you unplug from shore power?)
If that unit can do 4500 watts for 20 seconds it can likely do 3000 watts long enough to blow your fuse.
300 amps is the limit for mrbf fuses and that high amp fuseblock is probably also limited to 300 amps.
Overall, I'm not trying to squeeze every ounce of performance out of all the components, I just want something that can provide me enough power for my needs without catching my whole van on fire.
All my appliances running at the same time shouldn't exceed 150amps, and I'm ok with not running the PC and Pressure cooker at the same time to stay under 120a. Would I run into any issues by still using the 120a BMS while I look for a Daly 500a?