TBH, it actually sounds like the dog is an excuse. One or both of the parents don't want their child at your house. Maybe they don't want you alone with the child. But it sounds like every time you come up with a solution, they find another problem.
Has there been tension or disagreements, op? Could there be a reason for them not to trust you with the baby, or is the mum perhaps just not ready to let baby out of her sight, and you're asking too much?
I must say, 3 months is awfully young to be worried about whether he knows you or not.
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