We know McConnell is unamused by Trump's shutdown. McConnell said early on in Dec "there won't be a shutdown.:" Then he passes bipartisan spending to keep the thing open, and for Christmas, Trump shuts it down to please Rush and Coulter who theaten to rile up Trump's base.
What the dems really are doing is just putting the pressure on Gop senators. The public blames Trump .... because he's proud to own the shutdown, and he really did renege on his word to sign bills he'd agreed to. The Gop can reopen the gummit just by re-voting on what they already passed only with veto proof majorities.
McConnell doesn't want to piss off Trump, but he can let the Senate vote. And the gop leadership has no control over what Pelosi brings up for a vote