When a BESS supports both grid-interactive mode and island mode, then the required ATS-like device is called an MID, or microgrid interconnect device. The main difference from an ATS is that the BESS is always connected, so rather than switch to an alternate power source, the MID just disconnects the grid.
As for running HVAC or other large motor loads from BESS while islanded, there's obviously the issues of power and energy. Namely whether you have enough inverter power for the starting surge required, and then whether you can store enough energy to run the load an appreciable period of time, given the load's average power draw. Which for HVAC will depend on the outdoor/indoor temperature delta.
Having inverter driven (variable speed) HVAC equipment helps with both of those. More so with the power question, as I understand that the startup surge basically goes away, so you just need to size the inverters to the nameplate power rating. And somewhat with the energy question, as they are somewhat more efficient, so they move more BTUs with less kWhs.
The following information is specific to Tesla Powerwall, not Ford's F150 solution, but it discusses some of the limits, including running conventional (non-inverter) HVAC equipment off batteries:
Learn more about the variety of home appliances and EV charging equipment that Tesla Powerwall can backup.
www.tesla.com
Cheers, Wayne