When trying to send a teams invite from the shared mailbox (I do have full permissions) then there is no teams links generated
Can you share more details about how are you creating the teams invite?
For example, did you change the From field to use the email of the shared mailbox, or you configured the shared mailbox as an additional account?
I tested in my lab but seemed not able to reproduce this issue.
Below are the steps of a test in my lab:
1.my account has full access permission to the shared mailbox and it is auto-mapped in Outlook
2.create a meeting invite on the calendar of the share mailbox and click Teams meeting button to make it a Teams meeting
3.confirmed the meeting link, ID, etc. are correctly appended to the meeting invitation email
If it is still not working for you, please have a check if it is the same behavior in OWA following these steps:
1.login your account in https://outlook.office.com and select open another mailbox to open the shared mailbox
2.on calendar create a meeting and enable Teams meeting
3.send a test meeting invitation email and check if the url is appended
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