Canine Air Patrol?

[lordmonar]

I'm distracted enough when I have either one of my dogs in my car with me, I don't think putting them in a plane would be very helpful... as nice as they are.

+1....but the OP asked if there was any rule against it.

[raivo]

I can't help but think of the Connecticut wing patch...

[RiverAux]

Since the OP obviously wants to try to stretch regulations to take their dog, I don't feel any compunction about stretching regulations to say why it is not allowable and why they would face as much adverse action for doing so that I could gin up if I was the commander. 

1.  Transporting your dog is not official CAP business (60-1, 2-1-a).
2.  As an "occupant", you better figure out how to get that dog into the waist and shoulder harness (60-1, 2-1-g)
3.  The dog does not meet the requirements to be a "passenger" on board a CAP aircraft (60-1, 2-3) and won't be able to sign a release (60-1,2-3-b)

Keep in mind that CAP doesn't say that you have to be human to be an occupant or passenger, so you can certainly say that the rules apply to any living animal.

And most critically, I would dig up any other regulations I could regarding mis-use of corporate property that I could find and throw that in the mix and get the IG involved since this is certainly abuse and waste.

Why waste?  Carrying that dog causes more fuel/oil to be used than would otherwise be the case.  So, it may not amount to a lot of waste and abuse in purely monetary terms, but it is there.