When you elicit the ListPicker I expect the behavior of displaying the ListPicker selections behaves in a way similar to what happens when a user changes Screens and perhaps the Bluetooth client behaves as if you changed screens. I really don't know.
As an experiment, I suggest you try using a ListView component instead of the ListPicker. Try to select using the ListView (the selection process is similar but the ListPickers selection stays on the screen of origin) Use a ListView and see what happens. If you can select that way and cannot with the ListPicker, you just discovered an undesirable behavior of the ListPicker when used in conjunction with a Bluetooth client.
What do you discover?
Regards,