A conservation area does not make cooling impossible, it makes the outside of the building the problem.
The planning checker has the conservation area conditions in it, and the practical upshot is usually about where a unit may go rather than whether you may have one at all: the front elevation and anything visible from the road are where the restrictions bite hardest.
Worth saying out loud, because it gets forgotten: the things that need no permission at all are often the ones that work best. External shutters, shading, and stopping the sun reaching the glass in the first place do more for a south facing room than a machine fighting the heat after it has arrived. There is a piece on the site about exactly that.
If you have had a conservation officer say yes or no to something, saying which and why would help the next person more than any general advice.