Most remotes use rubber keys with a conductive paint on the back of the button. They make kits to restore the conductive backing.
I wish I could help. I owned a CP-45 for 20+ years and sold it last year. I was always worried about what to do if the remote failed. I searched around on the internet for replacement remotes (or codes) but gave up looking after a while. It is a very unusual remote unit. I hope you find a solution.
(I had the CP-45 paired with the CA-150 amp - great combo).
I hate a machine beating me, so I'd rather break it than not try to beat it.
I have successfully restored many (not all) remotes by taking them apart, giving all the parts a bath with a brush and alcohol, blow dry with compressor (or canned air), let thoroughly dry, back together ... peanut butter removed, hair, dust, oxidation.
back of the flexible sheets with buttons, the face of the circuit board(s). I've never had or tried to use the conductive paint mentioned above, the backs of the buttons have always looked good, but it's nice to know about it.
sometimes it doesn't work, but no worse off than before. If I scratch it cracking it apart, it's better than giving up, or not trying. even if you break a tab or two, they usually hold themselves together well enough.
Steve at VAS was able to fix my Cayin amp's remote, some functions worked, others didn't, cleaning didn't help.
Just a follow up - there are a few used Classe remotes on-line if you do a search, but some of the sites look a little sketchy.
The Logitech Harmony website indicates that their universal remote supports the CP-45. However, I think this is a discontinued device, but it is available as a refurbished model on Amazon.