Just a guess here, but since there's no room in F to stow a pet carrier in front of you, and pet's cannot be on your lap during take off and landing, you would be restricted to the Main Cabin where you could safely secure your pet under the seat in front of you.

I'll add that you couldn't be in Row 3 either, since there's no room under the bulkhead to stow the pet carrier.

On another airline, e.g. UA, the F seats do have under seat stowage so a pet in F would be allowed.