As a general rule, the use of the POP3 protocol is not a good idea (as I suspect Fastmail support gently advised you). With IMAP, there is no need for multiple separate connections to Fastmail. You can simply use multiple identities for the same account. Have you looked into that?
If you nevertheless want to treat the email addresses as though they belong to different accounts, you can do so by using the same main account and password you used the first time, but with an adjustment in the main account email address. For instance, if you originally signed up as
joebloggs@fastmail.com, you can use Joebloggs/fastmail.com as the user name for validation. Permitted names are
username@your.domain
username#your.domain
username/your.domain
username=your.domain
See
https://www.fastmail.help/hc/en-us/a.../1500000278342 for all the nitty gritty details.
The important point is that you do
not use the sending alias as your user name. You must still use the main account name and password.
However, just use IMAP and switch to using the excellent Thunderbird identity support.