question

I'd say the answer to this is "most likely".

We had a customer a couple of weeks ago with a similar setup (400Ah LiFePO4s, but otherwise very similar) whose Xantrex 3000VA inverter was faulting when his AC compressor started (also with a soft-start module installed). We bypassed his inverter and hooked his system up to our test bench MultiPlus 12/3000/120 and it did indeed run his setup with no problem, so I can definitely say that there's a very good chance that it will run yours.

I'd say a couple of uncertainties would be: the actual throughput of your lithiums (for example, Battle Born 100Ah drop-ins can put out an actual 100A, whereas the Victron 100Ah drop-in lithiums can only put out around 50A); whether or not your 360Ah of battery is quite enough to sustain that initial surge and then recover quickly enough to keep everything running smoothly; and, finally, your cabling... obviously you'll want to make certain that you've oversized all your cabling from the battery bank to the Multi to make sure that you're squeezing every last volt out of the system. When you're pushing a system like that, even if only for a few moments, a .75% voltage drop instead of a 1.3% voltage drop can be the difference between whether it works or not.

I'll also mention, just for the record, that a 300A alternator will certainly help you, but keep in mind that a 300A alternator is almost never putting out 300A... and indeed you don't want it to, because if you ran it at 300A all the time it would burn out within a day. I'd recommend looking into an alt-to-batt charger (Sterling has some great models like https://baymarinesupply.com/sterling-alternator-to-battery-charger-400-amp.html ) or a battery-to-battery charger (again, would recommend Sterling https://baymarinesupply.com/sterling-battery-to-battery-charger-12v-12v-120-amp-input.html )

...that's assuming you don't already have such a system in place to both protect your alternator from being overloaded as well as protect your lithiums from being over-charged.