"Anything but ...." means it could be anything, except for .... // basically, that the thing or person being described could be described as anything, but NOT that adjective. It is a way to say that the person/thing being described doesn't possess that quality
Some examples:
- He was anything but intelligent. = he wasn't intelligent at all, he could be anything, but not intelligent
- This evening is anything but pleasant. = this evening isn't pleasant at all, it can be many other things, but not pleasant
- I am anything but a quitter. = I may be a lot of things, but not a quitter
Your phrase "kindest regards sound anything but" means the regards aren't kind at all, they don't seem kind ... the full phrase would be "anything but kind" but "kind" is ommitted as it is already present in the word "kindest"
I hope this answered your question!