This is speculative, as I've not tried this, and I'm assuming that by 'the flavour of coconut milk' you mean 'coconut flavour'.
Creamed coconut/coconut butter, rather than coconut cream, may be your answer. It comes as a solid block, in solid form in individual sachets and jars.
Creamed coconut, also known as coconut butter, is a coconut product made from the unsweetened dehydrated fresh pulp of a mature coconut, ground to a semi-solid white creamy paste. It is sold in the form of a hard white block which can be stored at room temperature. https://en.wikipedia.org/wiki/Creamed_coconut
I use this rather than canned coconut milk or cream when making things like Rendang, because it doesn't have all the emulsifiers and thickeners of the brands I can buy locally, and at around 70% fats should incorporate more successfully into mayo than coconut milk which can be around 70% water.
When cooking you can just add solid chunks of the product to a sauce and it will melt and dissolve, but you can also melt it first and in this melted form you may be able to successfully whip it into a pre-made mayonnaise.
With the sachets you just place the sealed sachet into a jug of hot water until the contents liquefy, but you would also be able to melt some in a bain-marie as you would do with chocolate.
According to wikipedia
Coconut oil melts at around 24°C, so in warmer weather it is a liquid paste.
so you may need to do very little to get it to a state suitable for incorporating (I live in the north of Scotland so haven't experienced it in that state.)
I would suggest that to improve the chances of spreading the flavour evenly through your mayo, it would be best to add the mayonnaise to the creamed coconut a little at a time rather than dumping the coconut into the mayo. Having the mayo at room temperature rather than fridge-cold will also help avoid resetting the coconut cream before it is incorporated.
It is likely that quite a small amount would be needed as the flavour in this dehydrated product is more concentrated than in coconut milk, in my experience.
Edit to add: If it is just the flavour of coconut you are looking for, the other option might be a much more concentrated natural coconut extract.