Electing someone who won't give up their existing office

[Mr. Morden]

Suppose hypothetically that no one had won an electoral college majority in November of last year, with a bunch of Republican faithless electors voting for Paul Ryan.  So the House chooses between Clinton, Trump, and Ryan, while the Senate picks the VP from between Kaine and Pence (presumably going with Pence).

Now let's suppose that the House Republicans actually elect Ryan as president, but Ryan says that he actually doesn't want it.  He won't resign his House seat in order to take office as president.  Since it's illegal for him to hold both offices at once, I guess Pence then becomes president?  Would he take over for the full four year term, or could Ryan, at any moment, change his mind, and resign his House seat in order to become president?  Or would the House's election of Ryan, who's still in Congress and not quitting, be counted as invalid, and thus they'd be free to revote until someone gets a majority?

[Mr. Morden]

I thought that once the House vote was taken, Ryan would become president and automatically lose his seat. If he didn't want the presidency, then he could resign, but he would still no longer be a member of the House.

Really?  Because here's Wiki on the case law of the constitutional bar on holding two offices at once:

https://en.wikipedia.org/wiki/Ineligibility_Clause

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It sounds like because his eligibility was in dispute, he was not allowed to take his new seat in Congress until the eligibility question was resolved, which is different from saying that the election to Congress would automatically quit him out of his old job.  Presumably, it would work the same for POTUS as with Congress?