Being anti-car is more about being pro-infrastructure-that-does-not-rely-solely-on-everyone-needing-cars. It’s not even completely opposed to using cars/trucks for situations where nothing else will do.
My solution for this sort of arrangement would be to take the dogs and equipment for a walk down the street where a bus or light rail network would arrive every 2.5 minutes to take you to the train station. There, you play with or groom your dogs in a personal cabin on a bullet train that would take you to a similarily networked train station in the destination city.
Much of the required equipment would likely not be needed, since you could entertain your dogs on the trip rather than locking them away while you drive. If you still need a good deal of equipment, you could rent a car to get the station or hire a taxi cab.
Really unless you are making such trips daily, it wouldn’t make sense to need to own a car to transport a lot of stuff to the station in a nation with proper infrastructure. Just like you don’t buy a moving van because you plan to move at some point.
Out of curiosity, why do you take 10 dog crates for 4 dogs?
“Most consumers want fast food companies to label when sawdust has been added to food - but trust restaurants less when they do.”