Dropped Bluetooth Connection While Charging through Solar
A USB device requires that the input voltage be 5V (4.40-5.25V).  If the voltage is nonzero but outside that range, the device may malfunction or be damaged; certainly it can't be expected to charge its battery when input voltage is below spec. It is not appropriate to connect a solar panel directly to the USB input of a device because the voltage can't be guaranteed to fall in that range.  You really should have a buck-boost regulator between the solar panel and the USB device.

It wouldn't surprise me if the device failed to maintain a Bluetooth connection when the USB input voltage is out of spec. If it fails when the input voltage appears to be within spec, that suggests one of two possibilities:


The input voltage isn't really as stable as it appears. Check with oscilliscope on multiple time scales for noise or drops. Voltage drops can occur when there is a surge in current consumption by the device, such as when it transmits. A filter capacitor could be needed.
The device is damaged or broken. Check whether it can maintain the connection when powered from a normal USB charger.