Ah yes, the advertised bed length doesn't account for the space occupied by the scroll chuck, jacobs chuck and bit.  Been there - - -

When making canes on my lathe (nominal 18" bed length), my solution to this problem is to turn the canes in sections that I join together with glued loose tenon (ie, dowel) joints.  Each section is short enough to be turned and drilled, and also short enough that I can assemble two sections together and fit the combination between centers for final smoothing of the joint; I do this dry (without glue) so that I can do each of the joints before finally assembling the entire cane.  For strength, I make the dowels about half the final diameter of the cane.

A variation on this idea you might try is to make your tool handle in two sections with a mortise/tenon joint - drill the individual sections to receive the dowel/tenon and also for the hole that the tool shank fits into, and then glue the two sections together, and mount it between centers to finish and smooth the exterior of the joint.

Another idea:  It's not absolutely necessary to use a jacobs chuck to hold the drill bit.  If it is possible to mount the turning on a scroll chuck, you could drill the hole for the tool while holding a drill bit with vice grips.  The bit will naturally want to center itself, and if you are careful, you can get a hole that aligns with the turning axis.

I'm guessing you only have a pilot hole at best.
You could remove the tailstock and make a simple saddle or steady rest to support the handle for your method of drilling.

A drill press would also work 

You can make a mandrill and hand drill it. Take a piece of end grain wood, maybe 3 inches long. Drill a hole in it, about half way down that is the size of the end of your tool handle. While still centered on your drill press, drill a hole through it the size of your tool shank. Then put the big hole over the end of your took, and drill the tool shank hole from the other side. Not a perfect guide, but it does work. You can also mound a drill in the headstock, and maybe make a jig to keep the handle level and eyeball with the ways of your lathe and push it onto the drill bit. Both methods can work.

robo hippy