Wow, nice rig 2manybikes - and great shades on the pooch!
How about walking a dog on a leash and it tries to pull away? Does that mean don't walk the dog on a leash???? No...
You said your dog tries to get out but can't - that's different than saying he's not liking the ride and always trying to break out/free.
Have a great ride with your furry friend and your shiney fenders!