A trans man, gone through surgery, has a penis.
No, she doesn't. Back to biology class for you. A penis contains specific erectile tissues, like the corpus cavernosa. Have a diagram from Gray's to illustrate this. A transman might have a constructed phallus made out of tissue taken from the arm, leg, or torso. This is not a penis.
They should use the women's spaces?
Yes, or a single occupant unisex facility, or a mixed-sex facility. They might have to explain themselves in a women's space, but that's happened to me as well and it's not onerous. If you bothered to read my materialist feminism 101 post, you'd understand that capacity to undergo pregnancy versus capacity to force someone else to undergo pregnancy is the source of all sex inequality. Now, does this hypothetical transman fall into the "can be pregnant at some point in life" category or the "can rape someone into pregnancy" category?