hey! I don't think you need to change the filter pads at all in an axie tank.
The filter box probably assumes you are going to use it for fish, not axolotls. A healthy environment for fish is quite different from a helthy environment needed for axolotls. While proper cycling in nececary for both, i think it is much more nececary for an axolotl tank to have 0 ammonia because their skin is much more sensitive, being only a slimy membrane (it's so sensitive you are not ment to touch it, but it's hard not to sometimes with cleaning and feeding... I LOVE IT, IT FEELS SO SLIMY AND GOOD... without being too wierd. I definatley don't handle them)
the green/brown gunk that builds up in a cycled filter every few weeks is actually the good bacteria, the toxins in the tank are invisible and pass through the filter and get destroyed without notice. Cleaning should only be needed when blobs of poo or food get stuck in the filter media. Like you said, tank water should be used to clean it, and will not break the cycle, and replacing it all together will destroy the cycle. You do not need to spend the extra money, and even then, you do not need to fill it with specific pads designed for that filter and sold by the filter brand (rip off), you can buy a bag of filter wool, some filter sponge and some activated carbon and put it all in yourself! the only difference is it doesn't come in a little plastic cartridge. I find the cartridges leave a lot of empty space in the filter, when you could be stuffing it with more medium for bacteria to grow in. Don't pack it too tightly though, there needs to be room for water flow!