I know NOTHING about wood turning lathes, but for a metal working lathe, you can buy drills up to about 25 mm dia with an MT shank (No.1; No. 2, 3, etc, whatever you need). They are not cheap but I presume you only need one, and used sensibly, it'll last a lifetime and only need re-sharpening very seldom, if ever. And to further save money, and ideal for smaller lathes, you can get short series drills of this type, some call them blacksmiths drills.
I saw an add recently for Banggood (China - spelling?) who were offering a set of about 4 different dias for very little money, but no idea as to quality - or lead time.
But if either/both these points are important to you then try someone like Arc Eurotrade (their details are in the sticky at the top of the Metal Working Section here). Several others too who no doubt offer same/similar products.
I stress here I'm talking here about metal working drills, NOT brad point or anything else "specially woody", but especially for use in a lathe (well supported) and especially if a short drill (pretty stiff) my GUESS is that this type of drill will do you fine. The advantage of a suitable MT shank drill is you don't need a drill chuck (it plugs straight into the tailstock) and on a small lathe that saves length.
But DO double check with an expert, as said I know NIL about wood turning.
HTH