Addendum: all metallic alloys have higher resistivity (thermal and electrical) than either pure metal. For instance, sterling silver is more resistive than either silver or copper. I suppose aluminum alloys are fairly surprising in that regard, as they are still fairly conductive.
You could thermally epoxy one side if possible. Fine mating surfaces and compression during the face to face cure. Then build up a small bead of the epoxy around that fixed end, since those finely mated faces won't have much epoxy between them... at all. Silver filled IC chip epoxy is best for that, but many others are also suitable... if you decide to "fix" one side. Other wise, managing the two (four) faces that will be mating is possible, just a tad more difficult sometimes.
Aluminum alloys are poorer thermal conductors than pure Al is. Pure is like 210 W/m-K,and 6061-T6 is like 167 W/m-K.
Looks like the closer to pure, the better, but all searches nowadays point one toward Copper and alloys thereof.
'Ya know, if you can clamp the assembly quite tightly (or even apply some persuation as a part of assembly), you could use pure aluminum to fill the space, no grease. It's soft stuff, about like lead. Oh, and use springy washers too, I suppose...
The shiny things are copper heat spreaders, nickel plated. I did one DSRD-based  HV pulser that was built on a gold-plated copper block, water cooled; it looked great. You can get a peek of it here: