Currently I have several Account set-up in Thunderbird with the Matrix integration some more will be added soon.
It's embarrassing and may a privacy problem if you write a Mail but write it from the wrong Mail address.
Please allow to bind Mail and Chat account so certain user. So if I decide to send a Message to a certain contact Thunderbird auto set-up "From: " mail address according to the setup address.
In an advanced version, If I chose a certain user I had limited to a matrix account. The new Mail windows change to text-only and send the Message over Matrix.
To get it work properly there is a need to be able to setup per mail and chat-user of a contact in Address-book a corresponding mail address or chat-account of mine.
It would be nice If there is a statistic too in Thunderbird that could help determine the best message ways per contacts.
Sync it to CardDAV. As far I understand RFC:6352 you could create your own Entry.