Couldn't you do something a bit simpler with two drills rather than one at a carefully managed shallow angle?
The first is straight though the body, starting from where the strap button will go, running under the bridge and to the pickup cavity, the second is vertically, or angled, down from the bridge to meet that hole. The first hole will be hidden by the strap button. It's still a long thin hole, but doesn't need to be so carefully angled.