butane extraction is not the same as a supercritical fluid extraction, which would actually be fairly rapid. the pressure isn't high enough in a simple butane extraction.
if one were to employ an SFE device, then yes.. and basifying wouldn't be needed. A typical SFE process uses CO2, or a CO2/hexane mixture.
unless you're talking about a cleanup of a crude extraction residue, then this could be used as a re-x step, although DMT isn't very soluble in cold aliphatic solvents. this is why people warm hexane or heptane, prior to dissolving and evaporating.
if you can recover the product, then (safely) try it out.