I think Chester nn Chet works better on a person. Chet reminds me of the artist Chet Faker (not a bad association) and also Chet Hanks (kind of a bad association) so I like the separation Chester gives. At the same time I don’t love the double Er ending of your last name with Chester…hmm.
Maybe Winchester could work as a formal name? That feels a little more rugged than Chester on it’s own. And you could also use Win if you/he wanted later in life, too.
I was curious so I went searching and found this nameberry thread from a year ago that might offer some more options to you: