Had it not been for the Obamacare website screwup in 2013, the GOP would have suffered severely. The public blamed the GOP for the shutdown and the party's favorability numbers dropped 20 points.
I have no doubt, however, that neither McConnell nor Ryan will allow such a foolish outcome.
Your scenario is conjecture. Fact is they blamed the GOP because they were manipulated to do so by the MSM, because Boner and McConman had no PR strategy. The favorability polls were smokescreen - the bottom line is it didn't affect enough people to stick in the public's mind. They quickly forgot about it because there were at least a dozen or more Obama screwups closer to home floating out there to grab their attention.
I don't believe on taking a risk of shutting down the govt at the drop of a hat, but this is the time to force a veto, if Ryan will get his PR together and hammer it on the point that the Prez is risking nat'l security for the craven political agenda of gaining more votes for his party down the road via bringing in huge numbers of unvetted refugees and immigrants, it is a good risk to take.