"So if the election is tied = President Obama, chosen by the majority-Democrat HR."
You're assuming that it would be an absolute shoe in for Obama if that were to be the case.
In fact, NOTHING is a given.
It's happened several times in the past, and as with any action of Congress, significant wrangling and dealmaking can go on between supporters of candidates.
For example, the election of 1876 -- Hayes vs. Tilden.
Tilden won the popular vote, and had a lead, but not a majority, in the electoral college.
He won most of the states in the south, but in the end a deal was cut between Northern Republicans and Southern Democrats. For a promise of ending reconstruction in the south, enough southern Representatives came over to give the election to Hayes.