Hardware flow-control (CTS/RTS)
What you want is to be able to use other control lines (DTR / DSR). It's Data Set Ready / Data Terminal Ready, and it would solve your problem if your peripheral (or the BT module used) supported such behavior.

Are you able to add some HW at your side? Maybe you could implement a simple delay line by clocking incoming data through some shift registers, and waking up your MCU on incoming data so that (hopefully) it would be ready to receive it?