A pilot drill bit would work well for your task. A pilot drill bit will not wonder
if you can clamp the frame in a drill press. A pilot drill bit is much larger at
the shank and cut (ground) down to a fine point. They are available in a
large verity of sizes such as 1/4" shank with a 1/16th point.
https://www.google.com/search?source...drill&tbm=shop