In my experience, this works just fine. I live in a 4-bed detached, about 150 sqm, which has a 3.5kW indoor unit in the living room and a 2.5kW unit in the main bedroom. During the various heat waves this summer, the temperature reached 34C for hours on end at times outside. The A2A kept the house around 24C, with the rooms furthest from the indoor units reaching 26C. 

Downstairs, the rooms are in an L-shape around an integral double garage whereas upstairs the layout is somewhat more compact with the bedrooms and bathroom off a single corridor. As the warm air rises, the majority (about 60%) of the cooling was done by the unit in the main bedroom and we used a desk fan at floor level to push cooler air out of that room into the corridor / hallway. This was enough to reach the other bedrooms with the exception of the one on the north side of the house, which didn't heat up that much anyway.

The trick is to get ahead of the game in keeping the house cool. Close curtains on the sunny side of the house to reduce solar gain and run the A2A at relatively low power for a long time to hold the house at a cool temperature. That said, we came back from holiday in one of the heat waves and I could see that the room temperature was 30C when we set off on the 2 hour drive back, so I set the A2A running. By the time we got home the overall temperatures were well down and the rooms with the indoor units in them were already at 23-24C and had been for a long time.

My brother recently placed a deposit on a single split system install and it was recommended to him by the installer that he site the unit in the hallway and then open / close doors to achieve exactly the effect that you describe. He is going for a 3.5kW single-split on the installers advice.

I would recommend getting a few quotes and asking the installers for their advice on both sizing and placement. When you come to decide on which unit, take a look at the efficiency: SCOP for heating and SEER for cooling. These systems are highly efficient e.g. 5.2 on heating and 9.3 on cooling and so you will likely find it very cost effective to use the A2A for some space heating in winter as well as cooling in summer (assuming that you don't already have an A2W system). With A2A, you will be eligible for a heat pump tariff, 15p per unit and 5.2 efficiency is 2.9p per kWh of heat delivered which is about 30% the price of gas. Some of the cheaper A2A units aren't really designed for prolonged use heating as well as cooling, so I'd recommend spending a little more to get a higher quality one that will also deliver better efficiency as well as likely lasting longer.

@rob-of-york thank you. Hadn't occurred to me that some of the cheaper ones aren't that efficient for both heating and cooling.

i think a lot of people are going to have the same problem as me in that they realise they need cooling as well as heating and simply cannot afford to types of heat pump (I"ve established that air to water pumps are not good for cooling) and that means creating some kind of hybrid system.

My idea is to buy at most 2 air to air units (living and landing or bedroom) and hope they can do much of the heating in autumn/spring and am also wondering whether one storage heater in our large kitchen will fill the gap as I"m anticipating that this room might stay a bit cold and I could charge the storage heater overnight using our cheap EV tariff?  The other option could be something like the Panasonic RAC Solo but there isn't enough info on something like this yet.

I will look at this at a later date but I think people will need to be a lot more creative about the cheapest ways to heat/cool homes.

Long term, it may be more cost effective to get an A2A multi-split and have an indoor unit in the kitchen as well. 

The way we use A2A on a cheap overnight EV tariff is to run a higher target temperature overnight 22C rather than 20C and effectively use the thermal capacity of the house as a sort of storage heater. With good insulation, it takes a long time for the house to cool down and more heating to be needed. In autumn and spring, in fact that is all we need, the A2A doesn't run during the day at all and neither does our gas CH.

A2A is much cheaper to run and more flexible than an electric storage heater. Let's say overnight rates are 10p per unit and 30p during the day. With A2A at worst you would use about 50% of each, so an average price of 20p per unit. Combine that with a SCOP of 5 and its 4p per kWh of heating. By comparison, storage heaters are 10p per unit 2.5 times the price, and overnight rates are likely to increase as more and more people get EVs.

Whether it is worth putting A2A in your kitchen depends on the cost comparison with a storage heater and the number of kWh per year of heating you would use it for. For every 1000 kWh of heating, you would save about £60 per year by using A2A, or about £900 over the 15 year lifetime of the unit per 1000 kWh of heating. Which may shift things in favour of spending a bit more upfront.

You could go for a more comprehensive upgrade and get something like a Daikin Multi+ which has a single outdoor unit, 3 indoor units, and also provides hot water. The costs would obviously be higher, but if you replace an existing fossil fuel boiler, or direct electric heating with that, then you could potential be inline for the £2500 BUS grant. There are some hoops to jump through, mainly having sufficient heating in every room, but you can finesse that by first installing direct electric heating in all the rooms that you don't want to put A2A indoor units in. Providing the A2A can meet the whole house heat loss which is easily done, and all rooms have sufficient heating to cover their heat loss when existing direct electric heaters are also considered, then the system would be compliant with the grant. 

"The way we use A2A on a cheap overnight EV tariff is to run a higher target temperature overnight 22C rather than 20C and effectively use the thermal capacity of the house as a sort of storage heater. With good insulation, it takes a long time for the house to cool down and more heating to be needed."

That's really interesting and useful info.  If we manage to get it installed I will try that. Thank you

@mrkfm The Solo has a small capacity and only suitable for a typical bedroom or lounge, you'll also need a large hole cutter if not an installer! If installed professionally there isn't much difference in cost compared to say a 3.5kW single split (£1500) The single split is flexible where it is located and much quieter and efficient.