I haven't looked into the tooling much, but does it at least support pkcs11? That way you'd at least be able to store the key on a smart card or Yubikey.
Yes. Edit /etc/dkms/framework.conf, set mok_signing_key to something like "pkcs11:id=%01", and mok_certificate to point to a file containing the certificate. You can extract the certificate using eg "pkcs11-tool -r -y cert -d 01 > .../cert.der".
I don't know. I actually asked myself this very thing while typing the above comment, but I'm too busy/lazy to look it up.
One issue I can see with this, though, is that if the malware is already present on your system and can run things, nothing would prevent it from hijacking the modules or the boot image before they're signed.