so I guess i should make this a copy paste for the future as lately this has been coming up a lot. my entire cabin 800 square feet (roughly) and shop 1600 sq ft. was being run on one MS-PAE4448 inverter. thats a splitpack, a fridge, a chest freezer all lights an on demand gas hot water heater and the occasional microwave, heated table, hair dryer and of course shop use. (drill press, chop saws, circular saw, jigsaws, power planer etc, etc, etc.
Thats 4400 watts total or 2200 watts per leg 120/240 split phase. I could have went with the MS 4048 which is a 4kw single phase 120 inverter, but I knew that some of my tools work better on 240 and you can network (parallel) the PAE units for more power if needed, which you cannot for the non splitphase units. I added a more batteries and a second one for 8800 watts due to planned additions that pushed the single up to about 95% on occasion and wanted more headroom (WMO centrifuge that runs for hours on end).
the original setup with one inverter and 500 amp hours (27kWh) battery could run the split pack ac 24-7 on its own and when using wood heat in the winter months gave me about 2-3 days of autonomy year round while running the split pack to combat humidity in the house. (also important if you have fridge).
so in reality for a space as small as you are looking at a simple 3kw single phase inverter would be my go to in 48 volts. a seperate high quality solar charge controller, and one (1) DIY pack with 280ah cells would be enough for at least two probably three days while running the a/c. if you had 2kw of panels it would probably power your small house indefinitely probably sitting right around 80% most of the time and dipping occasionally down to 50% if you had a couple of bad days of solar production.
you have no need for anything bigger inverter or battery wise. insulate, get led bulbs, get a split pack and a small fridge and one last suggestion, get a small HF inverter like a samlex as your backup inverter something in the 300-600 watt range should power most fridges and you could still charge a laptop or phone at the same time. if you decide to expand int he future you are all ready at 48 volts and i have heard that the SRNE 3kw units can parallel if you needed more power.