Does it need a battery? If the bluetooth part transmits only every minute or 30 seconds you could probably get acceptable performance from a bank of capacitors.
A BLE module I have uses 15mA during transmit for around 1ms and sleeps the rest of the time. That's about 15uC for one transmission. Say you need to do 100 transmissions in every cooking session, excluding quiescent losses for now, that would mean you would need 1.5mC, or at 3V about 500uF. That's within the realm of a high temperature electrolytic, or even a few large format ceramic capacitors.