I am just spit balling here,
Could Carbon Nano Tubes bonded to a Graphiene base covering a resonance cavity constructed of copper/niobium or other material could be used to dissipate heat build up?
My idea is to coat the exterior of the super conducting resonance cavity with a bonded blanket heat sink then construct a cooling vessel around that layer that a pressurized cryogenic liquid would then be used to wick away heat buildup.
This article discusses the use of carbon nano tubes as a heat sink.
http://www.advancedsciencenews.com/nanotubes-on-graphene-make-the-ultimate-heat-sink/
The below article discusses grapheme coatings, while a flawless finish would give maximum surface area for the nano tube forest to be grown on I don't think it would be a deal breaker as we don't require a frictionless surface.
https://phys.org/news/2016-11-flawless-graphene-coatings.html