While just a little over your quoted price range (ain't that how it always seems ta go lol?) by far one of the strongest vintage looking yet new bikes available is Worksman's selection of "industrial" bicycles Worksman Cycles - Industrial Bicycles Home
While at first seemingly costly, another advantage with Worksman bikes is you needn't "upgrade" much if anything to make it engine-worthy as even the wheelset is quite stout with over sized spokes. Given that's often one of the first things replaced - the savings of not needing to drops it quite comfortably back into your stated price range.
Additionally, the Worksman is such an ideal platform that some of our members and vendors are offering custom made parts and accessories for it - including a fuel tank that's anything but "faux"
As for lookin' for an actual vintage ride? Patience... LOTS of patience lol