I don’t think I would bring Tater along, just for the reason that I doubt he will behave normally in a date after such a long drive (esp given that you know he stresses out after the vet). Plus having to make the round trip would be hard on him.
As for bringing a new bun home, I would line the carrier with some litter or a pee pad and fluffy hay, and the bun should do just fine. It also helps to put a rolled up towel under the carrier if needed to make sure it’s level (since most car seats are slanted), or put the carrier on the floor in the back seat.
When you get home with the new bun, it can also help to let them just sit in the carrier with no movement for 30 min or so. This seems to really help buns calm down in new spaces, as they feel safe in the carrier.
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The answers provided in this discussion are for general guideline purposes only. The information is not intended to diagnose or treat your pet. Seek the advice of your veterinarian or a qualified behaviorist.