Yes, that inner "pipe" should come out; I'm guessing
it unscrews. The outer "flanged" piece may unscrew too.
Doing it without damage may be a problem. I would think of
stuffing a deep socket inside the pipe to prevent it being
crushed while turning the outside with wrench/plier. Maybe
wrap with heavy cloth/plastic.

Yes, in my PF, the outer trim piece unscrews. This gives much
more room to get at the inner valve workings.

lee



Each mfr had distinct ways of doing things, thus the
dozens and dozens of faucet varieties.