Prince Philip was entombed in the Royal Vault of St George's Chapel untl Elizabeth died, then he was moved to join her in the King George VI Memorial Chapel. George VI himself spent several years in that vault before the chapel named after him was finished. My point is that moving remains can be done and often is for royals
They use special lead lined coffins though, which are ‘suitable’ for all that faff. They don’t bury them in cardboard boxes, dig them up and move them around. Huge difference.
Honestly, this whole thing is giving off Pet Sematary vibes. We have a much beloved household cat (is my adult child’s but it is the ‘family pet’ and I consider it to be my grandchild🤣), and when they die I honestly think we will all be in therapy as we love them so much, but still, this just seems unhealthy.
I can once had a heartbreaking conversation with parents. They went to a developing country for work, their older teenage child went and died there and was buried. They thought the work would be permanent, then after a decade or so were ‘reassigned’ and employer withdrew support. They wouldn’t leave their child so quit and became ‘stuck’ (had enough money that staying permanently was not a problem residency wise but at that point they wanted to return to home country). I recall cremation was not an option for them per religion. That’s the only time I’ve ever suggested exhuming as an option as they were so very unhappy and it was such a shit situation.
OP, personally, I think you should just move forward with your lovely memories. If you just can’t do this, then I’d have your cat cremated before being sent over as I really can’t imagine you will want to receive your beloved cat in the state it will be in after being exhumed, and that just seems so disrespectful to them.