Yes, pretty much any small split should do the job. There are loads available that are aimed at those with hot conservatories, and they are often cheap and around the right capacity to cool the whole of a well-insulated and sealed house. You can also get multi-splits, one external unit and several internal units, that may still end up being more cost effective that using an MVHR with a relatively low cooling capability. Our Genvex Premium 1L does not provide a massive cooling capability in practice, as, like all MVHR heating/cooling systems it is very limited by the duct airflow rates, so can only really trim the room temperatures a bit, not remove a few kW of heat from solar gain at all.
You could make a normal MVHR at least as effective as one with a heat pump in if a cooling/heating exchanger was added to the room air feed main duct.
These are water-fed, using chilled or heated water from the ASHP, so are easy to plumb in and control. There's no need to mess around with refrigerant, you just run a flow and return water pipe to the duct heat exchanger. If you fit a duct heat exchanger with a built in condensate tray (like this one: http://www.veab.com/documents/cwk/broschyr/CWK_VEAB_Heat_Tech_GB.pdf )
One of those duct coolers fitted into the main duct from a normal MVHR would provide the same sort of performance as the MVHR units with the built-in heat pumps, but at a lot lower price if you already have an ASHP.
FWIW, we find that by far the most effective way of getting rid of excess heat from solar gain is to just cool the floor slightly. I've been cooling our floor to around 19 deg C and that has a massive impact on keeping the house cool, far, far greater than using the Genvex in cooling mode. The reason it's so effective is that it absorbs a fair bit of the solar gain at source, from the sun trying to heat up areas of the floor, by swiftly moving that heat out before the floor has a chance to warm up and start warming up the house. It doesn't make the floor cold to walk, either, my other half tends to walk around on the travertine flooring in bare feet and reckons that at 19 deg C it doesn't feel at all uncomfortable.