You can add your disposable email addresses (or any email address) as a sender address for your account in Thunderbird, as described below:

From the Thunderbird main menu select Tools -> Account Settings.

Click on the account name for the account you want to add identities to. Then click the "Manage Identities..." button.

From there you can Add a new email address for that account. This address will then be available as a "From:" address when you compose a message in Thunderbird.