I thought early Babcocks in New Jersey sounded rare. At least not well represented online or in Stephen Babcock's Genealogy.
I can't say for sure if this will lead anywhere, but when I search "The Babcock Genealogy" by Stephen Babcock for the words "New Jersey" it points to an early Seventh Day Baptist (SDB) colony in New Jersey at Shrewsbury Church. Some of the next generation of Babcocks moved on to West Virginia and are pretty well documented, but it is possible some stayed in New Jersey and fell off the radar...maybe connecting to your branch.
Next I was looked for GED.MAtch connections on wikitree between Thomas Rouse Babcock Sr. one of the SDB migrants and those with GED.matches on Catherine, but nothing popped up. Still it is not a bad idea to scan for future Ged.Matches in any Babcock line associated with New Jersey by using the GED compare function.
Reading Thomas Rouse Babcock's profile tells a little of the small SDB migration to New Jersey.
The modern SDB church has some genealogic records and a big data base of their church records. And here's little overview of their migration where you will notice the "Babcock" name associated with Shrewsbury.
Wish I had something definite to send. Maybe you have already explored this angle.