“Tungsten carbide” as used in machine tools is actually fine tungsten carbide grains in a cobalt matrix.

Metallic tungsten can be deposited onto a conductor from a molten salt bath, at temperatures lower than the melting point of cobalt.
“Tungsten can be plated from a melt of sodium tungstate with a small addition of sodium pyrosulfate. Temperature 1650 °F” from http://www.finishing.com/270/89.shtml
also http://connection.ebscohost.com/c/articles/20308676/tungsten-galvanic-coatings-deposited-from-salt-melts

Carbide is reasonably conductive, and its COE range (4 to 7 E-6 /K typically) appears to overlap the value of tungsten (4.3E-6/K).
http://www.lucasmilhaupt.com/en-US/brazingfundamentals/coefficientsofthermalexpansionchart/

Carbide is pretty straightforward to form into shapes with precision of a few microns – this is off-the-shelf technology for making machine tools. And if it’s not conductive enough – well, given the voltages involved, the current should mainly flow through the tungsten skin… right?

If the solid-tungsten cathode is delayed, or if you want a spare, this might be worth looking into.