Tap Threaded Base Too Short To Secure - Options?

I'm trying to replace a broken worktop-mounted kitchen mono mixer tap - the new tap is secured by a ring attached to the threaded base of the tap body which fits through the mounting hole on the sink or counter (and not by the usual threaded rod).

Unfortunately the threaded section on the base isn't deep enough to pass through the depth of the worktop, so can't be secured.

Any ideas as to how I can remove a small depth of worktop underneath to allow it to be tightened?

Presumably the worktop has a hole in it already, so that a large flat wood bit wouldn't be able to centralise on anything?
This makes the job significantly more awkward but if you could fix a piece of timber - say 25mm thick - on to the top of the surface, you could use a hole saw to cut a counter bore on the underside....the pilot drill would centralise on the piece of timber above, the saw would cut a circular groove and then you could chisel out the remainder of the waste. Once the circular groove has started to cut the pilot drill can be done away with.
Even someone strong holding the timber piece down would probably work so long as the pilot drill didn't pierce his hands......!
Failing this, you could chisel out a sort of a counterbore from underneath which would have the same effect - it all depends on the access really!
John

Thanks, that was my thought too.

I think it might be easier to swap the tap and avoid potentially butchering the counter ...

That could solve your problems mate!
John