I think it would be a shame to not get a passport renewal for your other child, and to miss out on a family vacation, etc.
I live in Washington state. My divorce was amicable, and we did share custody and both travelled internationally many times separately with the kids.
We would get a notarized consent letter from the other parent, just in case. This was years ago, and maybe international parental abduction isn’t as much on the radar these days, but hopefully there is still some system in place to prevent a non-custodial parent from unauthorized travel with the kids?
I would also think that the US passport agency must have some way to prevent the reissue of a minor’s passport to a parent falsely reporting it as lost or stolen.
I can’t think of any VJ members to tag at the minute, but hopefully someone with direct and recent experience will come along. Best luck to you.