I have the same constraint. There are two options:
1. Drill in multiple bites. Drill as far as you can, then raise the table on the drill press, and drill further. A bit fiddly and it takes time, but it works.
2. Drill on the lathe. Some people have an almost religious affection for drilling on the lathe - for me, it's an option but not a panacea. To drill on the lathe, you need a chuck to hold the blank (they make special jaws for drilling, but standard pin jaws can be used as well), and you need a Jacobs chuck that you can mount on your tailstock (using a morse taper). However, while drilling on the lathe sounds nice, the fact is that the travel of the tailstock quill can be just as limiting as the travel of the quill of your drill press, and you may again be forced into doing multiple bits to get all the way through the blank.