How to ssh-copy-id 2nd key when the server only allows publickey authentication

ssh-copy-id appends keys to the remote authorized_keys file. To add several specific keys, run it once per key with -I <key-file-name>.

Update

After your comment, I think I got your question wrong.

You want to use one key to authenticate while installing another on your server.

ssh-copy-id does not offer a command-line option to choose a key for authenticating while executing ssh-copy-id.

But it passes -o through to ssh. So:

ssh-copy-id -i ~/.ssh/<your-new-id-to-install> -o 'IdentityFile ~/.ssh/<your-already-existing-id>' <servername>

You can also use ssh-agent, in which case you won't have to enter your passphrase when using ssh, scp, ssh-copy-id, a.s.o. for as long as ssh-agent is running.

Using ssh-add to add your key to the agent is definetly an option.
Otherwise I believe you'd have to set up an entry in the .ssh/config file which includes an IdentityFile option, this will tell ssh how to authenticate for that server and allow ssh-copy-id to do its thing.