Regardless of the specifics in the tax cut deal cut between the President and Republicans, the Democrats must allow it to pass and become law in order to save Obama's future creditbility.
Once a President says he's struck a deal, he must be able to carry through with it, or else he won't be able to be trusted in the future. Sure, almost anyone can bounce back from almost anything, but not that.
The deal can be tweaked in some small way, I'm sure, but basically Obama must hold to this agreement or the American people will question his leadership skills. Washington politics is generally about "negatives". Regardless whatever good the President has done, the American people will measure him by what is perceived as his failings and how he dealt with those. The key is to make those shortcomings appear much smaller and insignificant than other realities.
This fight between Congressional Democrats and the President is good though, because:
1. He gets to stare them down, and defeat them by passing the compromise.
2. The ire from the left ensures the deal doesn't drift further to the right, and maybe even gets marginally improved.
3. In future negotiations, Democrats have established that Republicans can't simply work with the President, because he, Obama, must work with the Democrats. It improves the negotiating posture of Obama in the future.