I’m confused does it have a leak or not, if it doesn’t have a leak it may be the coolant temperature sensor acting up or it could have air in the system still, these engines have a coolant bleeder screw on the top of the engine that needs to be open to bleed all the air out of the coolant system.
The purpose of the battery in the cartridge is to retain stored memory such as saved games so if you remove the battery for replacement or the battery dies it will delete the saved game. Sucks really hope this helps
The older cartridges should use the CR2025 the gameboy advanced cartridges sometimes use the CR1616, I buy my batteries off of amazon with the soldering tabs on them