I don't think the father was involved in the time travel contract, I actually think it was her aunt.
Her aunt is a spirit caster, she knows that she can see spirits, so the aunt knows that mc is the real deal. She probably got worried about what was happening with the Cosette situation and set up the contract as a safety measure.
This however does not explain why daddy dearest has been having nightmares, but then again, he's from a family with magic powers, perhaps a powerful memory can linger through time. I count losing your real child and regretting it, as a powerful memory 😅
Anyway, here's another series added to my list of things I read, thank you for the chapter!