Firstly, it should be noted that the Manchester encoding library which I am using receives messages in the background with the aid of a regular Timer 2 based interrupt. This should mean that I can receive readings even when the Arduino is receiving/processing/sending serial data.
No, it does not mean that. Serial data processing requires interrupts, too. It is possible that the interrupts are interfering with each other.
This suggests to me that there is something about an active Bluetooth connection which is interfering with the reception of sensor readings.
You mean that one radio interferes with the other? Quite possible.