I fly in the USAF and this is something that alot of aviators go through. If it works on a person it may work on the dog. Dramamine/Ginger root didn't help - the only thing that did was to constantly keep flying and the body will adapt to it. Air movement and being able to see ie. if your dog box has a side that gets wind and allows the dog to look outside should help.
that aint no lie bout being able or not being able to see. iv got over 600hrs over 400 under goggles, logged in a Chinook, and 100hrs or so as a pax and didn't get sick. but every stinkin time I get in a 130 or the german 123, I get sick to my stomach, start getting hot flashes n wanna throw up. its happened a couple times on large ga acft in sever turbulence as a pax. its happened a couple times though as a student pilot, especially with flicker vertigo. I had to hand the controls over the my cfi 1 time bc it was so bad, I couldn't see, had tunnel vision so bad, my vision went black/blue, saw stars, nauseated, but aftr a min or less, I was fine, just had to chug a lil water to keep from filling the cockpit floor with my lunch. I hope my luck holds out when I get my commercial rotary wing add-on