Omnicharge is smart, but sometimes it can be too smart. To save standby power consumption, Omnicharge's Auto Shut Off function will turn off the Omnicharge when the current drawn is lower than a designated threshold. The problem with your Bluetooth headset (or other small gadgets) is that it has a small battery inside and only requires very small current to charge.
You can disable this feature by choosing NEVER in the 'Auto Shut-off' menu:
a) Double press the Power Button entering the menu, and choose AUTO SHUT OFF
b) Choose NEVER
NOTE: Remember to turn the 'Auto Shut-off' function back on afterward so that your Omnicharge won't consume power when in standby mode.