The Democrats want an amnesty bill or no bill at all.
I think you are correct that, politically, the BEST place for the GOP to be would be to have a popular bill that they passed, ... OR a popular bill the Democrats loudly filibuster.
Instead, the GOP in the Senate is trying to bend over backwards to get a 'bipartisan' compromise. Big mistake. The Democrats are wnating a bad bill for all the wrong reasons.
After all this time, they still think the Dems will play nice, GOP ain't called the stupid party for nothing.