They could pass a separate bill on guest workers, i.e., more slots for skilled labor. This would be a sop to the Chamber. I am dead set against it, but it would smoke out the Dems.
And if they want to be too clever by half, they could support a bill that legalizes the status of the illegals once all of the enforcement measures are in place for a number of years and shown to be operating and working effectively. This would be effective PR for the Reps. The public wants tougher enforcement so it would be difficult to attack by the Dems in the midterms. It could be called a "trust but verify" strategy recalling Reagan and his dealings with the Soviets.
"....legalizes the status of the illegals once all of the enforcement measures are in place for a number of years and shown to be operating and working effectively."
The phrase that comes to mind is "playing with fire." Every "comprehensive immigration" bill has something like that, and you see how well that has worked so far. Imagine the most extreme pro-amnesty interpretation about the "enforcement working effectively" that the 9th circuit, for example, could conjure up. Then Roberts or Kennedy could chicken out, or President Hillary (voters don't seem to hold Dems accountable now) could replace the older guys on the Scotus.