If I were in the Republican leadership in that state, I would hold daily press conferences, pointing out the Dems' cowardice, and how they're holding the state hostage.
Nope. The republican should just open the legislature and call for a voice vote that a quorum was present, then vote the law into being. Congress does this kind of crap all the time. Someone actually has to object and force a roll call, but that would require a dem be present, which would create the quorum. From what I understand that's all they need for the quorum, is one single dem, so the dems can't even just have a ringer there to force proper parlementary procedure.
This isn't rocket science. When Dems act like children you just have to go on without them.