Do you have access to a clamp meter, so that you can measure the actual amps to the inverter under the high load? Also it may be helpful to check amp draw at each battery, to see if they are each contributing roughly the same amount. (Those draws are likely going to be fine though.)
A 250a fuse is a little high for the 1/0 wire. If you replace the inverter wires with 4/0, you would be good. (Note that your inverter can only make use of a max sustained 215 amps [about 2200 max watts / 12v / 0.85 efficiency ~~ 215a].)
As far as harming the batteries, no worries as long as they are healthy, and your amps rate is less than the battery's discharge rating. E.g. if each battery is rated at 75a sustained discharge rate, and you are drawing 200a with your load, you're drawing approximately 50a per battery. But, if a battery fails, then the other 3 are going to approach 67a each. And if one of those remaining 3 is "aging" faster than the others, and/or there is a connection problem, then conceivably you could exceed 75a on 2 of the 3 remaining batteries.
It's a nuisance, but for peace of mind you can separate the batteries once in a while and do a simple voltage and load test on each one. That would verify that it is still contributing well for (at least short-duration) high loads.