Red Herring. Who gets branded with it? GOP or Dems.
If/when this bill gets dumped, you can be assured the Dems are going to shout from the rooftops that "the American people wanted immigration fixed but the Republicans, caving into pressures from big business and right-wing zealots and racists, opted to do nothing." Count on it, it's going to happen.
The only way I can think of to counter this is to have a border security only bill to offer up immediately following the current bill's demise. This way it puts the ball right back into the Dem's court: the American people want the borders secured...are you Dems for it or against it?