A Counterbore may be the way to go here.
Stewmac have them, not cheap.
https://www.stewmac.com/luthier-tool...g-reamers.html

A step drill might also work if you are careful.
The small one out of these sets.
https://www.bunnings.com.au/craftrig...l-set_p0027584
https://www.totaltools.com.au/power-...it-individuals
https://www.ebay.com.au/itm/3PCS-Con...cAAOSwTu9f4rLf

Or maybe try to find the exact size in this type.
https://www.ebay.com.au/itm/Twist-St....c101195.m1851
https://www.ebay.com.au/itm/Twist-St...AAAOSwPvZgHJOn
https://www.ebay.com.au/itm/Hexagon-....c101195.m1851