I don't know if the House will pass the Senate bill as-is. The entirety of the bill can't be passed via reconciliation (and what does get passed will be temporary and shoddy). They can finalize the compromise bill and vote before Brown is seated, but there are political problems there with senators like Jim Webb who have already voiced opposition to such an idea. That leaves...
...The nuclear option.
Roll back the filibuster to 57 votes -- or just revisit the Harkin-Lieberman plan.
The plan he announced with Lieberman 14 years ago would have slowly scaled down the cloture threshold for legislation that had been filibustered. The first vote would require 60. If it failed to reach 60, debate would continue until a new vote, which would require 57, and so on until a simple majority could determine whether the measure lived or died.
The big catch is to be ready with a strong counterpunch to the wingnuts who will totally flip their shit when this happens. It won't be a very popular decision in the near-term, but it'll be much better for the Democrats than not passing any bill. Failing to pass healthcare reform now would incite doom in November and years of Republican majority nonsense in Congress.
Adding... I meant to note that even if they use reconciliation or shove something through before Brown is seated, that leaves an entire agenda to be addressed. How else can the Democrats repeal DOMA or pass financial reform or climate legislation without 60?