Oversimplifying somewhat, there are two basic reasons for the White House and the Democratic Congressional leadership to be open to accepting ideas/requests for changes from individual or groups of Congressional Republicans to a bill they are trying to move.

The first is that a Republican member might have a good idea, one that improves the bill substantively, from a policy standpoint.

The second is to draw Republican votes so as to be able to claim whatever enhanced legitimacy and public support may (or may not) come from having Republican votes.

Where the point of accepting a particular Republican request to alter the stimulus package was to try to win a Republican vote, but the White House thought it made the bill worse, then accepting the change was a "concession" that should not have been made in the first place. Now that the Republican bad-faith MO has been revealed, such "concessions" should not be made in similar situations in the future.

With Congressional Republicans who are now showing their MO to be as negative as the Republican minorities in Congress in 1993 were with Bill Clinton, then the way the White House should think about the benefit of being bipartisan is solely for the first reason above--to be open to ideas that would improve a bill and policy, and not to be able to say they had bipartisan support for something they did to try to obtain greater legitimacy.

If they think a particular Republican-offered idea improves the bill, they should take that suggestion. Just don't imagine it will draw any Republican votes--unless they secure a formal commitment that that member or the group asking for the change is "on" the bill--will vote for it--if the change is accepted. 

Because if they take a Republican change that they didn't really think was a good idea, with the hope of getting some support from across the aisle, and then that member or group of members votes no on the bill anyway, where are they?  They've made the bill and policy worse, with no benefit to show for that.