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While I don't know about how Thunderbird works in this regard, I do know that pgp/gpg encryption also verifies the sender against a known key.
This means that the cretificate you created is most probably tied to your primary email.
So you'll probably need to set up a new encryption certificate specificlly for your alias.
Yep this is the official way. There is also the quick and dirty way of adding a new ID to your existing key to enable its use with a different email sender, but be wary of the info leaks this could lead to…