My perception of Sting is that he is a seriously erudite and intelligent man who has made a large amount of seriously popular and well-written music.
That said, he may feel that the idle gossip of a tea-time tv show is rather beneath him.
So he should either bite the bullet and accept the interview opportunities his people set up for him, and do them with grace and professional courtesy, or he should ask that he doesn't do them.
Freddie Mercury gave next to no interviews, he didn't like them, he certainly didn't need them, so he didn't put himself in the circus ring - Sting should do the same.
I have interviewed hundreds of musicians, lots of them on Sting's level, and above, and almost without exception, they are polite and professional, and the couple of exceptions - literally two in thirty-seven years, stand out.
Sting needs to understand that he does no-one any favours to turn up on national television with a sulk on.