The cost of the batteries would offset the cost of the inverter though. I know some of the units are marketed as not needing batteries, but I don't accept the premise that these heatpumps are going to be effective when shutting down and starting up every time a cloud goes by, not to mention what it would do to compressor/motor longevity.
An APSystems QS1 inverter is $400 and will run 4 panels for a total output of 1200 watts, that cost would buy 2-3 12 volt batteries. I'd still be going grid tied.
I think there are some misconceptions. PV is not on off when a cloud goes over the sun so the minisplit would probably not turn off, it just may not be cooling. A mini split is not like a conventional window shaker or even standard home heat pump. A window unit is generally a single speed motor hooked to a compressor. The compressor is either on or off so there is lots of current inrush everytime the motor compressor starts. It runs at 100% until it overshoots the temperature setting and then the compressor turns off. The fans may have multiple speeds but they do not pull much power. A mini split maintains one temp by running the compressor at variable speed adjusting the compressor speed as needed. Even if it does turn off, its a "soft start" where the motor is ramped up slowly so it will last a lot longer than one that cycled on/off. Thus my guess is if the power from the panels is not high enough the fans may still run but the compressor does not. Conventional old style variable speed drives convert the input AC to DC and then converts it to AC waveform. So having a 48 volt DC input may make the drive simpler without a front end AC stage. Industrially most motors are now VF drives and as long as they are built for VF drive use they are lasting longer and far more efficient as they are soft start and run at lower speeds most of the time.
I think a big selling point of the DC units that do not require battery's is that they are black out and brown out proof, given the issues with the grid in Texas where the legislature is more interested in bailing out the corporations than consumers, black outs and brown outs are likelihood in hot spells. Lot to be said for something that runs for cooling as long as the sun is out. Its not just Texas, CA has issues especially rural areas that get their power shut off to keep from the power lines starting fires. In both areas AC is not optional, people die especially the elderly when the AC goes out.