Hipshot makes tuners that don't require any screw holes at all to be drilled, and in fact, will cover up the old screw holes. They use the so called UMP (Universal Mounting Plate). The tuners come in sets (6 in-line (no help in this case), and 3X3), sealed and open back (sealed ate 16:1 ratio, open back 18:1). there are a dozen different button options, and in locking and non-locking versions. I used a set on a Squier Strat (open back locking tuners, 6 in-line), and they are very smooth, very stable and well made. They are made in Korea.
In your case, you would want 3X3s, non-locking, in whatever finish (Chrome, Nickel, Gold, or Black), with a button of your choice, and either open or sealed (the open tuners are a little lighter). They are significantly cheaper than most other good quality tuners. They do need 10 mm. post holes.