Thank you!
This solved the problem! Now I can use Thunderbird to send email when specific from address is needed! 
…however it would be nice to have this kind of feature at the webmail too…
And I don’t know why it is not allowed, as long as domain part of address is domain what that axigen installation is serving.
Because user can always create alias and that point it’s allowed to send mail with that address… It would be easier, if you don’t need to create alias every time…