Modern refrigerators are VFD based, so theoretically they should run on DC. If not, remove the PFC module.
I haven't seen any refrigerators in the US yet that have a variable speed compressor, got any examples? Mini-split air conditioners are pretty much the only place I've seen inverter based compressor capacity control. Well, that and commercial >100 ton chillers.
Most capacity control which is now becoming commonplace for air conditioning or refrigeration is based on cylinder unloading (reciprocating) or partial scroll bypass valves for scroll compressors. My parents got a Trane system installed a couple years ago which has a 2 stage scroll compressor, 1.5 and 3 tons of capacity respectively.
I was able to do some preliminary testing with some equipment I had laying around. (VFD connected to mains) They are all single phase 240v devices.
Neslab CFT-33 laboratory chiller
Capacity: 950 watts or ~3200btu/hr
Compressor: CSIR type with current based start relay which engages a start capacitor
Pump: positive displacement 1/3hp, has centrifugal start switch, RSIR type
Reservoir: Plastic, 1.5" pipe cap access
Notes: compressor does start and run with sufficient power but due to the start relay the compressor is stalled until approx 30hz and ~7A of current which engages the start capacitor.
Unknown manufacturer mini chiller
Capacity: approx 100 watts or ~315btu/hr
Compressor: RSIR type with PTC thermistor starting device
Pump: positive displacement, 1/4hp, has run capacitor, PSC type
Reservoir: Stainles steel, large panel access
Notes: compressor starts easily at low speed but once started must cool down for several minutes before the PTC device will allow starting again