3-way switch for light with fan controlled by switch built into fan. Odd wiring layout. Wiring assistance required
I think I get what you're describing. You need to get switched power from the living 3-ways back through the common box and bathroom box on to the fan/light in the living room, all the while powering bath and dining lights as well as the fan, all with only 14-3 cables between boxes.

I think where you are going wrong is supplying power to bath and dining using the red conductors from the common box, because the black is for the living room power. You don't need to do this, they are all on the same circuit. Use black as common power to all switches and fan. The red is then free to be used to run switched power through all the boxes to the living light. And except for the switched white, all neutral whites tie together. Do not ever connect switched power white to neutral whites. Instant short circuit once any switch is on!

In the dining/living 3way switches box, black from common box to common of BOTH 3-way switches. Travelers OK as connected. Switched living white to red from common box.

In common box, connect all like colors together.

In bath box, connect all like colors together. Pigtail from black to bath switch.

In living room. Black to fan power. Red (switched) to light power. White to common neutral.