No, if democrats vote no the Republicans will vote no as well. A loss in the house would be devastating for the dems. It would show the democrats and independents that even the democrats don't believe this garbage.  Going to the senate to have a week of people pleading the fifth and more bickering would just please Republicans. Independents would likely just hold the continued theater in contempt.  I think what Lindsey is doing is the right call.  Edit: just think about all the good senate work and judge acceptance that they would be missing out on.