One other item to look at... Is the chuck attached to a mounting plate? Typically on smaller lathes the three (and four) jaw chucks are screwed to a mounting plate that is threaded to go on the spindle. Ideally you want to take this mounting plate, mount it to the spindle (screw it on) turn it true to the spindle, clamp the chuck to it and indicate it to get it centered on the plate and running true on the spindle then take the whole unit off and drill your mounting screws.
IF it were mine I would but a drill bit in the chuck backwards so most of the shank is pointing out, use a 3/8" or bigger if you have it. Indicate off this. This should tell you, as others have indicated, if you have run out (the chuck is not true to the spindle). If this is the case and it, the chuck, has a mounting plate you should be able to remove the screws, follow the procedure outlined above and rotate the chuck on the plate to allow you to redrill and tap the holes. I have had to do this on a couple of 3 jaw chucks over the years cause I bought them after getting the lathe and they did not run true.
Just a thought...
DonT