So just for arguements sake. If he was Hewitts son and not Charles would he still be a prince due to Diana at the time being the princes or is it purely birth line?
Im assuming it is birth line.
If that was actually the case (and it was known publicly) then he'd be an illegitimate son of Diana and wouldn't be a Prince/Royal. He's a Prince because he's a grandchild of the Monarch via a male line, any kids of Charles, Andrew and Edward are automatically entitled to be Prince/Princess (and styled HRH) though in the case of Edward they chose for their kids not to use the "HRH" titles and they're styled instead as a Son and Daughter of an Earl - since Edward has an Earldom. In the case of Anne, it wasn't automatic for her kids though AFAIK she wouldn't have wanted it, her husband(s) would have been offered Earldoms when they married her too, if accepted then their kids would be styled like Edwards kids are - "Viscount" & "Lady".
A few generations ago Great Grandchildren via a male line would also be Princes/Princesses albeit "HH" rather than "HRH" (though those whose grandparent then becomes Monarch would become HRH). These days it is only William's kids who get Prince/Princess and "HRH" titles.
So, for example, Harry's kid would have been Prince/"HH" back in the day but under current rules he isn't, he is, however (as the son of a duke) entitled to use one of his father's lesser titles as a courtesy title - so could be an Earl or a Baron, but they've chosen to just use "Master" instead (at least for the moment, he's still due to inherit the Dukedom when Harry dies).
Back in the day though Archie would be "HH Prince Archie" and once Charles is on the thrown he'd be "HRH Prince Archie" as the grandson via a male line.