Child support obligations are not discharable in bankruptcy, so S2's bankruptcy is irrelevant, unless your goal is to get a lift of the automatic stay so as to foreclose a judgment lien for past-due child support.
However, if the divorce judgment did not specify any support amount, then S2 is not in arrears, i.e., he owes no support at this time, and it is not a "core function" of bankruptcy proceedings (at least in my view) for a bankruptcy judge to liquidate S2's ability to contribute to the minor children's support to a number and then enforce that number against S2's property in bankruptcy.
Frankly, based on your post, it would appear that S2 has no obligation to pay one cent, until such time as a family court makes a definite and certaint child support order against him.
So, now, your turn -- what am I missing here?
Basically, you want to hold up the bankruptcy court while you go get a child support order. But, at the moment, there isn't an order, and the BK court can't give your new order retroactive priority to the date before the BK filing (at least, I can't imagine how that would work).
Or, maybe you want to ask the bankruptcy court to rule that the divorce judgment claims some "equitable" amount of support, and therefore the court will liquidate that amount in fairness to the child.
In either case, you would be opening up the federal courthouse to making child support determinations, and a flood of new litigation. I think the judge will trot out the abstention doctrine and send you packing.
The solution to S1's problem is to go to the family court and get a real child support order and hold S2 to it. And, if S2 doesn't pay, then S1 files for contempt and gets a money judgment for arrears and then executes/garnishes S2 and his property, and/or has S2 doing community service/jail time for failing to pay support or seek work, etc.
Note: I do admire your zeal, but I think it's a waste of $10 for the parking garage.
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