GREETINGS HUMANZ
I've run across some forum posts with people lamenting the difficulty in replacing the backup battery on gameboy flash carts once they go. Mine died after about 4 years of use, and I found it quite easy to replace. With that in mind, here's a completely unnecessary How-to for replacing your dead flash cart batteries. The flash cart I'm using here is an f2a Ultra 2GB.
Step 1: Find your flash cart, and put it within arm's reach.
Step 2: Using your thumb, or some small prying tool, gently separate the front plate from the back. The PCB inside is not affixed to either side, so be prepared for it to fall free if you are holding the cartridge above a shark tank or something.
Step 3: You will find the battery (a large silver disc) setting just right of center on the front. The two contacts for the battery on my cart were attached to the battery in a way that would not allow for easy removal. To get around this, I used a small knife to gently separate the metal contacts from the battery, starting from the edges. You'll find that the contacts are only really stuck in two places on each side, so with some patience and a gently touch, the battery can be removed. Be careful not to wiggle the contacts too much, as they are only connected to the PCB by a solder blob.
Step 4: Inventory your parts and make sure you haven't somehow lost something in the last two minutes. Displayed here are the battery (a CR-2025 3v Li disc), the PCB, and the case pieces. From this point, you are ready to replace the battery. Just find a replacement (the CR-2025 is a very common watch battery, check K-mart etc) and affix it to the battery contacts on the PCB. The side with the CR-2025 designation should be face up, and make sure you have left the contacts oriented such that you don't reverse them when replacing the battery. One thing to note here, is that these batteries typically last 3-6 years. It's not unheard of to use your cartridge for longer than that. With that in mind, try to use some means of reconnecting the contacts that will be more maintainable than the way it was when you cracked it open. Personally, I chose electrical tape, as it was laying nearby and can easily be removed when I'm doing this again during the next presidential election. You could also use a conductive adhesive, solder, or magic - just keep in mind that you might want to remove the new battery when it dies!
Anyway, 90% of the information here is superfluous, but I thought it worth explicitly mentioning that replacing the backup batteries on flash carts is neither impossible, nor technically challenging.
Enjoy saving your LSDJ data again! 
