Personally, I would not remove any more wood than absolutely necessary from a Fluke or Flea headstock.
The string tension is pulling the tuners against the front and back of the headstock, that with the stock Grover 2B friction tuners has about 6.5mm of wood on each side of the hole, and if you narrow this too much, you are likely to have issues (maybe only on tenor scale) with higher tension strings causing the wood to snap.
I've installed Grover 9NB machine heads that can use the same factory hole as the original friction tuners, as well as have installed the Gotoh UPT-L tuners, on several of my Fluke and Flea ukes. the Gotoh UPT-L tuners require the existing 6.5mm tuner hole to bored slightly larger with a reamer or unibit to about 10mm, and even so, only about half-way into the headstock
All of which is detailed in the Fluke and Flea mega-thread which you can get to from clicking the first link in my signature below. Also within that thread are links to the threads where I have installed these other tuners on Fluke and Flea ukes, with photos, of course.
As far as the countersink from the INSIDE of the headstock (the site of the string post), not sure how you would do that, and with the 'friction' part of the tuner with the button turning against the wood itself, it seems like it will both wear down the wood enough and possibly apply enough force in the tightening of the tuner to cause the thinner opposing sides (front/back) of the headstock wood to just split from the force or pressure of having the tuner tightened down.
I may be too paranoid about breakage here, but we all have our own tolerance for risk.
If it were me, the Waverly tuners are like $50 on StewMac, and the Goto UPT-L tuners are like $60-70 from HMS, I'd get the Gotoh instead, which are planetary geared tuners with a 4:1 ratio, a lot like banjo tuners, and very light weight.