DC current surge suppression
The inverter may or may not have an inrush, the refrigerator certainly will. They will typically have a vicious starting current, sometimes even 10x the nominal running current. 

There's not a lot you do about this at the fridge motor. Most folks simply oversize the inverter and its supplies sufficiently.

As the fridge inrush is mostly reactive, some smart people build a two-stage inverter. They use an initial DC-DC converter, sized correctly for running, and then a DC-AC H-bridge capable of handling the awful power factor.

As your inverter is over-sized sufficiently, it should be enough to handle the surge between your batteries and inverter. As you've found, 30A is almost enough. Could you bypass the battery to inverter components while the fridge is starting up, perhaps with a beefy FET and a circuit that senses the start of the current surge?

While you could build a current-limiter, this would probably cause your fridge to fail to start.