Mumble has a url handler that is installed by default with the standard msi installer. The format is mumble://[USERNAME]:[PASSWORD]@[SERVER ADDRESS]:[PORT]/[CHANNEL]/[SUBCHANNEL]/[SUBSUBCHANNEL]/.../?version=1.2.0 (see http://mumble.sourceforge.net/Mumble_URL).
I propose a new output line for the picked players private message notification adding the line mumble://mumble.atf2.org:64738/TF2/[SERVER]/?version=1.2.0 where SERVER = da0, da1, etc.
The ?version=1.2.0 is necessary for now.