AP grants the customizer as part of the starting items. You can access the customizer tab from the buildgun and use it, and the bottom right shortcut hints show it, but pressing the keybind does nothing.
Perhaps something extra is needed to actually unlock the keybind? Check with vanilla content that unlocks it too?
https://discord.com/channels/731205301247803413/1018853131859267656/1447957793020837958