I’ve had a carrot cat too! I sliced some up and dehydrated them then sewed them into a toy and she just loved it! The trick I found is to break the pieces inside the toy now and then to release more of the smell (same way catnip works really).
Does your carrot cat also like swimwear that’s been used in a chlorinated pool, but not yet washed? Mine treated that like catnip, too.
I used to have a cat named gin, but not named after the alcohol.
Her name was 銀 (gin, pronounced geen like green) which is silver in Japanese.
She passed this time last year, quite unexpectedly, after 16 years with me, but your post brought some happy memories :)