Doesn't look like it's removable - at least we couldn't get a faucet
seat wrench in there - it looks pretty round - not sure if there is
something deeper inside that a wrench would click with - but we got a
wrench with 3-square and 3-hex options and none went through. So we got
a dressing tool and gave that a shot.

Still leaks.

I think it may be time to call a plumber, we've been messing with this
for months now, to no avail.