In that scenario, are FPUs a good option? Are there other drawbacks?
Even then, there is a fair limit in cooling and efficiency you can get from them because of the lack of phase changes occurring.
If you have a balanced mechanical ventilation (both inlet and outlet) that would be one of the best options to condition the whole house without putting air conditioning units everywhere. Having said that, we turn on the air conditioning in the attic during hot days which removes the biggest heat and moisture from the house. Next up is installing more sun blinds to prevent the heat from entering in the first place.
Insulate all piping properly, the outdoor part will freeze in winter, and the indoor part will drip in summer if you don't. The best products for this are foamed rubber based. Make the insulation seamless with the right glue or tape, every opening will drip.
Add manual or automatic valves to shut off the heating loops in summer, so you don't accidentally get cold water in there. Your cold water loop might also need an extra circulation pump, depending on how you run your piping, but that's hard to say without a lot more detail.
FCUs are available in the same form factors as indoor units for refrigerant-based systems, including a thermostat remote control, e.g. https://www.sinclair-solutions.com/files/user-manual-en/fan-... . But get one from the manufacturer of your outdoor unit, so the communication lines work, e.g. so the indoor unit can instruct the outdoor unit to start/stop/regulate.