AC is a must only if you really using it for 2-3 months of the year cooling wise.

AC would've been the best and only option vs mvhr as you are asking, if you didn't already had the HP, as it would've ticked heating and cooling, without hot water.

HP cooling it's not equal to AC cooling, but who says cooling with HP it's a bad idea, they are bad at their job.

So given the fact that you already have the HP, and that the UK period of "must have" cooling, it's actually very short, you will do just fine. You will have to be mindful of not running the pipes too cold to avoid the condensation, and pre cool the system so it heats up as the sun shines, if you let the house become hot hot, then a hp cannot cool it fast like an AC.

Some people due to lack of budget do just fine with Ceramic single room Heat Recovery Ventilation, but I don't know your house or your quote. Keep in mind you actually don't need a high flow in real world, so maybe you can look for options/smaller cheaper mvhr system. The real life need for air exchange is smaller than people think, but it does improve quality of life. 

So do whole house mvhr/localized heat recovery depending on the budget, and learn how to use the HP for cooling.

Couple of things: First, have you considered getting a decentralised (i.e. single-room) MVHR just for e.g. the bathrooms and kitchen? They will be cheaper to install, but obviously don't provide extraction and fresh air for the whole house, just the rooms where most moisture is generated. Essentially they're extractor fans with heat recovery built in. Second, have you considered getting AC only in, for example, your WFH office(s) and/or master bedroom? I.e. the rooms you might need it most? Basically if you scale back a bit and are ok with the compromises, perhaps you can do both.

Having said that, if I were renovating and stripping everything back anyway, I'd go with MVHR as it will provide most benefit in this country and, as you say, it's harder to retrofit. Then when you have the cash put AC in the rooms you want.

@cashback would you mind me asking what "AC" option are you looking at?

I already have the ASHP and would like to put cooling-capable chilled water driven high wall or ceiling FCUs into a couple of rooms. just those that overheat massively (2 bedrooms) and my office . I'd be running a separate new circuit to them to avoid condensation issues. But having difficult finding appropriate emitters. hence the question.

rads+ufh+fcu from the same heat pump shouldn't be impossible and is not a fundamentally bad idea. Rads and ufh that is the one that costs you some efficiency - higher LWT  probably needed for the rads then blend it down cooler for the UFH, but if your ashp designer is good they should be able to figure out a away to build it.

@iancalderbank have you tried these?

https://www.oceanairuk.com/product/daikin-fwt-ct-2-pipe-wall-mounted-fan-coil-unit-2-43kw-5-28kw/

A late reply; but given the extreme heat occurrences the UK has been having, a solution for over-heating rooms because of sunlight is external shading. At the high tech end there are adjustable fins on a metal frame; better still are deciduous trees providing shade, external wooden shutters, or in extreme conditions hang a sheet or cloth over the window outside. I used hessian sacks in the very hot (40 +) days.

Thought I'd better chime in with where we got to on this.

We ended up going down the MVHR route. The system is installed but not yet commissioned (pending a number of other jobs; like I say, it's a major rennovation).

The reason the heat pump for cooling was deemed a bad idea was for many reasons. Firstly, space - we'd need a "cold battery", i.e. a large volume of water that the heat pump would cool, which would take up a lot of space where we don't really have the space, then be pumped off to fan coils round the house. My house is an odd shape, so the pipe runs are very long, and at risk of condensation, as well as the tricky spots to run condensate drains for the fan coils themselves. While it's true that there are other condensation concerns with MVHR, all the air in the ducts is warm, running through cool(er) spaces, but the semi-rigid pipes are insulated and under insulation, so the risk is minimal.

That said, we are actively exploring underfloor cooling using the underfloor heating loops.

The other major reason we chose MVHR is that the air quality in our area is really bad; our house is on a busy road that suffers a lot with stationary traffic at peak times, and there was already evidence of soot in the corners of a lot of the rooms at the front of the house, and around air bricks.

The MVHR unit itself is on a north facing wall that is always noticabley cooler, so it should yield a reasonable cooling effect in the summer months at the expense of some loss of efficiency in winter.

The house will be reasonably air-tight and we are doing everything we can to ensure air tightness to places like the loft. Possibly not EnerPHit standards but "good enough" for the system to work effectively. The install itself is fairly unobtrusive and constrained to the loft, with all of the ducting drops for the first floor hidden in a void between two adjacent and opposing wardrobes, or in a false wall at the end of the bathroom that already hides plumbing.

We have a MVHR, installed 13years ago and we think it’s brilliant. No long lasting cooking smells when you don’t want to open the windows in winter. No road dirt (the filters get filthy and we used to breathe that) No heat loss through trickle vents. You need to ensure that toilets have a boost and overrun setting which comes on with the light.

We dry clothes inside and never have condensation problems not even in the cold corners (poor house thermally).

For summer cooling we have wind-out canopies over the large south facing windows and the neighbours have big trees. There is a 6degC drop in temperature from hot to pleasant when the kitchen canopy is wound out. No on-going power consumption with this shading approach either.

Good luck with it