You can buy smaller ones. Domestic heat pumps usually start around 6-8 kW for single phase ones and go up to ~20kw for 3-phase units. Keep in mind that these values are the peak power of the heat pump, which you would rarely use. Modern units have inverter motor driven compressors, which can modulate their power output in the range 25%-100%. 100% is not a happy place for the heat pump if it is cold (and humid) outside, because it will quickly accumulate ice on the outdoor unit and start doing many defrost cycles per hour, which lowers the COP (efficiency).

With respect to the duty cycle, obviously if you have solar power, you would prefer to use it predominantly and only add up some extra power from the grid when needed. This is the essence of the sizing problem, because that leads you to 2-3x power overprovisioning and the need for heat/cold storage. Heat storage can be two types - DHW and space heating. Space heating is the easiest to estimate. You need to know your house's heat loss (either by specification or just figure it out empirically if you have already lived in it). DHW storage is more difficult to estimate, because it depends on the usage (e.g. how many showers per day). Cold storage is the most problematic, because the fluid needs to be at least 16C or lower to do useful cooling work, however you cannot go much lower than 7C unless you are using propylene glycol (expensive) and even then your indoor units may start to freeze (I am not even mentioning indoor humidity management and dew points).

Lately, the industry has been exploring PCMs (phase change materials). The idea is to store heat/cold not as sensible heat, but as latent heat of the phase change. In practice the substances used are either salts (efficient, but corrosive to the storage tank) or paraffins (more expensive, less efficient, but still viable). These come rated at a specific temperature, but usually have some hysteresis/drift and other issues. I guess you are now feeling a bit frustrated from the engineering complexity :). If batteries were cheap, long lasting and environmentally friendly, this complexity would not be needed. However, I really doubt it that in the foreseeable future batteries will beat heat storage. Given that most of our domestic energy use is space heating/cooling and DHW, I think that PCMs may actually have some moat. There are already offerings on the market, but IMHO they are still not very compelling. What I see lacking is some integrated offering, that would take into account the PV schedule and also grid prices. One a side not, batteries still have an advantage if you can sell back to the grid at a high premium or if you need to e.g. charge your car in the night. So these technologies may be complementary, rather than competitive.

A very big factor is climate. Just to give you an example, I live in the mountain with a colder climate. Cold water from the faucet is around 10C. I rarely need cooling if at all, but I need space heating around 8-9 months during the year. Just 300 km south and by the sea (Greece), cold water from the faucet is around 20-25C, you need 4-5 months of cooling and only ~4 months of heating. Some countries, such as UK have very moderate climate without extremes and things are more predictable. Where I live, we get -15C in the winter and 38C in the summer.