you only need an extractor set, cost about £8 or thereabouts at most.
the little hole in the middle of the adaptor will be enough to get a purchase on and undo it.
taking the head off is making a massive job of something simple.
and yes, go from underneath.
if there is some sticking out, then blutack in the hole (to stop matal going into oil feed) and chiesel the remaining bit of the fitting round so it unscrews.