Interesting question/need.
The only idea I have is to have a common master address database which
in turn is then used by the different (mail) applications.
It might also be possible to "combine" the different back ends into one
front end. But not without major programming efforts, as far as I can
judge. Curious what other people think.
However, in a first effort, I'd try to get rid of the "Collected
Addresses" data in Thunderbird by merging them into my Personal
Addressbook and accordingly changing the destination under Preferences >
Composition > Addressing - "Automatically add ..."
But it's only a workaround.
Nino