It's completely unclear from above your own situation and that of spouse, ie who would sponsor, are you compliant with residency rules, why your minor son wasn't included in previous application, etc.
But assuming all the rest of that makes sense: your US citizen son does not need a visa to enter. If PR process for him is not finished before, you can apply for an extension of his 'visitor status', preferably 30 days or so before the six month stay period is done. Even if you don't get a formal response/approval of the extension request, he will be fine and allowed to stay while it's being processed. (Applying 30 days before advised in case it's returned for error which would then leave him out of status or the implied status from the extension request)
He should not therefore need to exit and re-enter.