Well,the main problem is with travel time (cca 8 hours).
Maybe do it in two stages? Allow an hour between trains so you can take a stroll and your dog can take care of its business.
I can't reserve whole coupe,well i could,but i should probably buy all six tickets which is nosense.
Just travel off peak (middle of the day) and go first class: I've never been in a first-class compartment which has more than a couple of people in it. If the people in your compartment really object, just move to another one and keep going until nobody in the compartment objects. Worst comes to the worst you can always sit in the corridor. Or in the toilet.