what is the best scope mount for the handi's using existing screw holes?
Well now.. what caliber barrel are you looking to scope? I ask because you added screw diameter as a parameter. I haven't seen much if a need for larger screws on handi. Although, my 500, 454 cassull and heavy loaded 45-70 sure could likely benefit from some #8s...
Back when I was big on Contenders. I had sheared off its screws on a 357 Herrett. I tapped it for number 8"s and sheared those!!! I found some better read harder screws and combined with epoxy never had an issue with the barrel again. Then again, it also needed double rings to keep the scope from moving under recoil too!!
I also agree with Long Tom and like the factory base combined with Leupold or Maxima rings.
CW