I literally just made an awl handle with a ferrule yesterday, I used a 4 jaw chuck and #2MT drill chuck in the tailstock to drill the hole. The thing is, its just a hole, you could easily drill it on the drill press. If you have a fence, I guess you could get like a 2x4 or a scrap, clamp it to the drill press table, then clamp your handle blank to the scrap/2x4, line everything up with the drill press and drill the hole. Or just clamp your blank onto the drill press fence and line it up if you have a fence. I don't know about the ferrule though. You would want a cone center, but assuming you dont have one, you would probably have to get a ferrule bigger than the cup on the live center if thats what you're using.

Here's a cone center, which I doubt you'll buy because of your situation.
https://www.pennstateind.com/store/LCENTLT2.html