The right 3-way (center switch) has neither hot nor neutral. The three wires present there are:
- messenger 1
- messenger 2
Since you have neither hot nor neutral, that far 3-way is dead as far as feeding power off it. You have to go back to the left 3-way.
You also have a problem there paralleling two 12-2's, despite what the illustration shows, you are not allowed to do that.
I would start the whole thing over with smart switches which use radio waves for 3-waying, and simply straight-line hot and neutral through all boxes.
- Switch-1: Smart-switch master that supports radio comm with a powered (or battery) remote.
- From switch 1 to lamp 1, black=always-hot, white=always-neutral, red=switched-hot.
- Inside lamp 1: TOP 12-2 is passthru: black goes to always-hot, white goes to always-neutral. BOTTOM 12-2: Black goes to switched-hot (red wire) and to lamp. White goes to neutral and to lamp.
- From lamp 1 to 2: TOP 12-2: passthru hot and neutral. BOTTOM 12-2: switched-hot and neutral for lamp 2 only.**
- Inside lamp 2: TOP 12-2 passes through. BOTTOM 12-2: black is switched-hot to lamp. White is always-neutral to lamp only, do not parallel with top 12-2.
- From lamp 2 to switch 2: black=always-hot, white=always-neutral, red=capped at both ends.
- Switch 2: Appropriate smart-switch remote.
- From switch 2 to switch 3: Black=always-hot, white=always-neutral.
- Switch 3 and past: normal switch circuit.
** This arrangement is very important because it assures currents are equal in all cables. You can't randomly parallel two 12-2's if you need 4 wires, the diagram you linked is wrong.