I’ve done this before on a Pokémon silver cartridge. Instead of soldering the tab back on, I carefully taped it in place. It works just fine. The fit of the board into the plastic case might be different on the advanced cartridges so I’d say use your best judgment but it will work if it’s just held there instead of soldered.

To remove the soldered on tab, I just used a tiny screwdriver (like for eye glasses) and removed it.

The nice part with this is you can just change the battery if it goes dead again.

Working on power supplies is dangerous because they have giant capacitors that store a LOT of energy that can discharge very quickly. It can kill you. You need equipment to safely discharge the capacitor and need to know what precautions to take but often it isn’t worth the hassle and risk.

Working on a small battery like this isn’t very dangerous. The real risk is ruining the cart because of amateur soldering. If you are comfortable with your soldering skills then go for it.

related question:

Let’s say you have a saved game on a pokemon game cartridge
You know the battery will run out soon, and you want to replace it

However, the instant you remove the battery while the game is off, your data gets wiped.

Would it (theoretically) work to replace the battery while the game is running? Since the memory is being powered by the normal gameboy batteries (cylindrical double AA ones)