Running Serial, POE, and HDbaseT Through a Patch Panel

The correct answer is you don't "fix" it at the patch panel. If you follow the code and it doesn't work, you need to fix the other side.

One of the reasons people use T568A/B and not any straight through wiring pattern they want is that it makes troubleshooting and repairs much simpler, as you are finding out now. If you don't fix the other end now, you will be in the same situation (or someone else will) 5 years from now while trying to fix the camera side of the cable.

I also firmly believe that infrastructure cabling should be infrastructure cabling, not a "long patch cord" running through a building. So I would terminate the camera end of the cables to a jack and use a patch cable to connect the cameras.

If you really must take the shortcut (for instance, it is more important to get the camera up ASAP), then you can do so, but put this on your priority list of things to fix ASAP. Just "reverse" where you punch down, since brown is where white/orange should be on the far end, punch it down where white/orange should be. White/brown to orange, green to white/green, and white/blue to blue.

I also recommend that you develop a cable standards document detailing how you expect cabling to be run for your organization and provide it to any contractor working at your site. Then enforce the document before you allow them to get paid.

Splitting a 4-pair UTP cable is completely non-standard, especially with PoE. You will never get a split cable to pass the category suite for that cable category (in your case, Category 6). If you want to run gigabit ethernet over the UTP cable, you will need all four pairs to do that.

The correct way to do this is to run a minimum of two UTP cables to each faceplate. Your WAP is probably going to be in the ceiling or on a wall, and it should have its own dedicated cable, in its own cable pathway.