No. True fillibusters no longer exist, due to rules changes by the Democrats when they had the majority. They are not 24/7 nonstop droning sessions. Now, filibusters are these virtual, weakened things that do not really have the potential to cause the PR damage you are hoping for.
If it's no longer a 24x7 requirement, then why was Frist bringing in cots to sleep in?
With aid from the constituents of the RINOs and the rest of us, Frist can restore the original filibuster and additionally make the instigator of a filibuster remain on the Senate floor for the duration of it. Institute a rule requiring continuous speech with no pause of more than three minutes duration, disallowing the yielding of the floor to members of one's own party, and no points of order during the filibuster.
If it hurts to do it, one would have to have a passionate belief that it is necessary. Then if a member can last long enough to affect the opinions of a majority of the whole Senate, so be it.
Fine, then why not honor the institution and traditions of the Senate by removing those rules changes? Force anyone who wants to filibuster to do it like "Mr Smith Goes to Washington". How can that be spinned negatively, it is in the finest traditions of the Senate. Then when the dems try to fillibuster in the future, they have to really do it!