You can do it either way as long as both of you agree. You can use TurboTax to see the tax implications (if any) of claiming it one way or the other.
You can file Married Filing Separately and divide everything, including the exclusion in half, which is $250,000 for Married Filing Separately, if you qualify qualify for the exclusion.
Please see Is the money I made from a home sale taxable?
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