What kind of inverter do I need for an inductive load?
It's easier to teach a violinist the bongoes than the other way 'round.  

You should be shopping for an A/C unit with "soft start".   This is one of the reasons for these units to exist - ability to start off inverters, generators etc.  They use an inverter drive to run the compressor motor.  They are also able to run the motor at different speeds to suit demand, so they don't do the slam-on/slam-off thing like simpler units do.  

This platform doesn't allow product recommendations, but I would be asking on the off-grid/solar forums where people deal with that all day everyday.