Should the spouse earning more and paying the mortgage get to claim the house on their tax return?

Does it really matter? I'm being serious. The determination of who claims the mortgage interest (which is what I assume you mean by "claim the house") in their itemized deductions should be based on whatever is most beneficial for the couple... whatever lets the two of them considered together pay less taxes and retain more of their money. Generally, because the income tax is a progressive tax, it turns out that this will, in fact, be whichever of the two had a higher adjusted gross income.

Except in very special cases, if you're worried about this sort of thing, the "good" news is that it's probably the last year you'll have to worry: you'll likely be divorced before the issue comes up again.