Agree with the sentiment of not using metric threads on an Imperial machine., only a source of confusion in the future for you or someone else.
Maybe I've misunderstood, but having used a 5.5mm drill, a) M6 would be a loose fit (Tapping drill should be 5mm) and 1/4 BSF will be 6,354mm OD anyway.
If a 1/4 BSF bottoming tap won't clean the threads sufficiently, my advice would be, if enough meat is present, to drill and tap to 3/8 (BSF or BSW). If you drill out in small increments, the drill should follow the original hole.
Make a bush with 3/8 external thread and 1/4 BSF internal and use an anaerobic sealant to lock it in place. As a fitting tool, put a nut on a 1/4 BSF setscrew and run the bush into place, flush with the surface, back off the nut and remove the setscrew. Once the anaerobic has cured, (allow 30 minutes minimum) reassemble (It may be worthwhile to grease the new eccentric screw, so that the thread is contaminated, and any stray anaerobic does not cure on to it) and you should be back in business.
Howard