If your private key has a good passphrase and is suitably encrypted, say with ed25519, then that's probably as good as you can do other than physically going into work and storing everything in your head :-) Politely ask the client to suggest what they consider would be a suitable alternative.
I also setup git hooks to prevent accidentally checking in private keys or passwords into git or other version control systems. And if I'm travelling into or from work I also encrypt some stuff just in case I have a problem and the laptop is stolen.