The other alternative is like current state of the art 3d printing ceramics - you either replace some high percentage of the filament with clay and fire it as a post processing step and it burns off the plastic, or print a clay/water slurry directly and fire it after drying.
But I don't think we'd end up with the basalt being very filamentous.
If the binder that gives you something printable at low temperature doesn't integrate into the final result through chemical reaction, you are almost assuredly going to get a high porosity mess where the binder had to vaporize out.
If instead the binder and precursor can melt, react, and expand into a solid that precipitates out because of a super high melting point, the expansion will ensure that you get a fully dense part that can be machined back down.
But I don't think we'd end up with the basalt being very filamentous.