those older bikes can be quite power hungry, so it would be better to get a battery that can provide at least 25 amps continuous.
The battery and motor would be happy with 36v because the voltage of a lithium battery is approximately the same as a lead one. the controller regulates the power to the motor, so you can't overload the motor by using a different type of battery.
There's a lot of room in your battery case, so you have lots of options for filling it with lithium.
Personally, I wouldn't use LiFePO4. It's much heavier than other lithium types and generally has a lower power potential, so you need a bigger heavier battery to make up for its deficiencies.
The obvious question is how much you want to spend and whether you have the ability to construct your own battery?