AFAIK, Jimmy, it’s not a question of grounding the pup and pots to the bridge – you need to ground the strings to the amp to stop them being great, long, receivers of extraneous noise. The only other way I can think of is to have a strap dangling down from a right-across string tree, which would look a bit naff.

Sorry, can’t think of an alternative to drilling the hole 🙁