Pc power supply Ac in rush current overloading inverter
I don't know how to solve your AC inrush problem. But it may not be the only way to approach this.

The most common problem with inverters is that people underestimate how large the DC wiring needs to be. It is very possible that during startup, the DC current is high enough to cause excessive drop in the wires leading from 12V battery to inverter. If so, this drop could cause the inverter to fault out due to low DC input voltage. That is the first thing to check, since you have test equipment and knowledge of how to use it. Check the droop at the input terminals first. If it droops a lot during changeover, then work your way back to the battery. It is possible that the cabling is adequate, and the battery is weak or undersized. Measuring will help you sort it out.

If that doesn't fix it, another option is to use the inverter strictly as an inverter, and run the power supply from the inverter at all times. Buy a battery charger to keep the battery topped off from the grid. With this setup, there will never be any inrush. This is what I have in an outbuilding. I have an 80Ah battery serving a 35W AC load through a 600W inverter. An intelligent battery charger keeps the battery topped off at all times. It is worth getting a good inverter, because it will be running 24/7.