Will that still be a valid signature? If not, how can I detect it before it goes its rounds in our systems?
It’s still a valid AES (Advanced Electronic Signature) in eIDAS terms, which is what is valid in Denmark, as long as it’s clearly marked that the name is a pseudonym and not simply a false name. The resulting signature will still contain the user’s PID number (and CPR number, if configured), so that the signer can be uniquely identified. So yes, the signature is legally valid.
You might want to treat these signatures as special cases for your business processes, however. There is only one way to do this, unfortunately, and that is to see if the signer’s name exactly matches the string “Pseudonym”.