Strikes me as a Bluetooth limitation. It only connects one pair of devices at a time so a handoff is not really feasible.
If each copy of the app was updated with the history of connections, the app could possibly tell the app what to look for but it will scan for any open connections and it’s really the Bluetooth at the device and not Dash and that manages the connection. The phone app isn’t aware of what other devices are available since it is a single point to point protocol that is occupied while connected to the Dash. It doesn’t know whether the computer is in range, has Bluetooth turned on or whether the computer Bluetooth is already occupied with another speaker, headset ot other device.
If you use a telephone call as an analog to Bluetooth, if I’m talking to you, I can’t pass you on to my brother. You have to call him or vice versa after we hang up. Bluetooth doesn’t have party lines; you can’t connect your phone to two differenr headsets at the same time.