So the top 4 screws will not be an issue at all, the counter sunk angled screws are just performing like a lag screw principle so in all likelihood they are insetting right at the heel into the neck substance.
The amount of shim you will need to get a perfect action will be minimal from what you say, you could take the neck off, mess about with a couple of playing cards to get the amount of shim height, tapering angle etc you need before slipping a nice veneer in there and do it right. Drilling through a bit of wood so thin for that job is lemon squeeze.
May just make it just perfect for you, no more will it won’t it hassles