would i be within my rights?

http://www.azleg.state.az.us/FormatDocument.asp?inDoc=/ars/28/00939.htm&Title=28&DocType=ARS

Sounds to me like I can shave my taillights completely and just run a 3rd brake light that acts as a running light too...

No, it says on the rear of the vehicle. The cab would be the middle, and it also has to be able to signal flash left and right.

Also, how would the guy behind you know where to stop in the dark when he sees your light on the cab and can't see the bed?

No mike by law the light has to be at the rear most part of the truck. I know this fro ma few people that got tickets just running third brake lights like you wanting to do.

Also getting rid of the tail lights could cause problems with the law that talks about having red reflectors on the back of your vehicle. For example, clear tail lights are technically legal, however removing the stock lamp [and therefore the red reflector] is illegal.