BeO is nearly abandoned because of toxicity. Al2O3, AlN, (BN) are common. Mica is older.
Do you have a minimum thickness, like a breakdown voltage or a maximum capacitance? If not, a thin foil is the solution.
Also: good contact is more important than the material itself. Use special grease, or at least some oil, but no dry contact. Some elastomer films provide the good contact naturally.
All this is well-known from power electronics engineering. Look in catalogues from Berquist and the like.
-----
Breakdown voltage or maximum capacitance impose a limit to the heat conductivity, once you've selected the best possible material. If you have to go beyond, a solid won't fit. Blow some gas, or use a heat pipe (with an insulating wall, sure).
-----
Just because someone wrote "price is not an issue": diamond isn't the ultimate material. If you remove 13C and synthesize diamond from pure 12C, you improve thermal conductivity a lot, even at room temperature.