There is one error in your reasoning.
I assume that they are married. You say that if they file separately, the spouse who owns the property will claim itemized deductions with the mortgage interest and property taxes while the other spouse will use the standard deduction, and together they will end up paying less tax than if they filed jointly.
This scenario is not allowed by the IRS. When filing separately, both spouses MUST use the same deduction method (either both Standard or both itemized deductions). If one spouse use itemized deductions, the other must also itemize even if the itemized deductions are much lower than the standard deduction.
This has nothing to do with the fact that one spouse is an immigrant. In fact, a green card holder is a resident alien and is treated exactly as a US citizen.
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