I wonder if Tony Malykh's bluetooth audio addon might help. It is intended to stop the computer's blue tooth from going into standby for audio purposes, but I don't see why it wouldn't work for your Braille device's bluetooth connection too.
This is the web link:
And this, copied from the addon page, explains how it works:
"Most bluetooth devices enter standby mode after a few seconds of inactivity. That means that when NVDA starts speaking again, the first split second of sound will be lost. Bluetooth Audio add-on prevents bluetooth devices from entering standby mode by constantly playing a silent sound, that is inaudible to a human ear."
Hope it helps :)