How to fix this bathroom light

Contact your local building department, and ask them if this would require a permit, and if you can do the work yourself. That's the only way to know for sure, as different areas have different rules.

You'll likely have to pay a small fee for the permit, and have the work inspected at different stages of the job (or maybe only once it's done). Most areas let homeowners do the work, as long as it's inspected.

If you do unpermitted work in your home, and something does happen. Your insurance claim could be denied, and/or you could be held personally liable for injuries and damages to others and their property.

If you're going to use wiring similar to the first diagram from that thread, and you're planning to use 12 AWG wiring. You're going to need five conductors (plus ground) between the switch boxes, so I'd probably use two separate cables.

In the case that the power originates at the fan/light, I'd make use of some 12/2/2. Using a length between the fan/light, and first group of switches. Then another length between the two groups of switches.

NOTES:

• Grounding conductors have been excluded from the diagram, to make it a bit clearer. Make sure all grounding conductors are properly installed.
• Notice that the white wire that's being used as a traveller between the switches, has been reidentified at both ends using a bit of black tape/marker/paint.
• As @Wolf Harper points out, make sure the two cables between the boxes stay in close proximity with each other. Or use conduit with single conductors pulled through it.