Replacement battery voltage mismatch tolerance?
If the battery you want to use is a typical lithium ion battery then what you want to do could be very dangerous.

Rechargeable lithium batteries require carefully designed charging circuits to prevent them from being overcharged with too much current or a voltage that is too high. A difference in charging voltage of 0.65 V could destroy the battery or cause it to catch fire.

Whether or not a battery with a voltage higher than 3.6 V would damage your device is impossible to say without a detailed knowledge of the device itself. Someone could guess, or give you an opinion, but we can't give you a definite answer based on the information you have provided.