Clint is going to propose to Kim, with the proposition that as a married couple the two of them could fight for custody of Stacy's baby.
Kim allegedly takes Clint up on his proposal (of course.)

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If this is true, then short of Clint having money to bribe the judge, I am not getting how the two of them (married or not) would have any palpable legal claim on the child.

This is true, Gigi as next of kin or Schuyler or Fish as the baby's daddy (whenever they establish this paternity) would have more of a claim then Kim and Clint.