OK, given you are deciding to do fish-in cycling (which I am OK with, since it is the safer option for your axolotls given your cats), you are going to need to figure out what levels of ammonia and nitrite are the highest that are safe for your axolotls, so you know when to do water changes.
Here is the annoying thing - you need to allow some ammonia and nitrite to build up, so the bacteria have food to grow and reproduce, but if too much builds up, it will harm you axolotls.
How much ammonia you can let build up depends on the pH of your water, so you will need the test kit to find out.
Keep nitrite under 0.25 ppm. You probably won't see any nitrite yet - you need to get enough of ammonia-eating bacteria growing first before you will start seeing nitrite.
I've never used products that are supposed to add the right bacteria. Many people say they don't work and some people say they worked great for them. I suspect it depends how old the product is, among other things - bacteria are alive and need to eat, and perhaps they starve in the bottle. There has been some debate as to whether these products are safe, but the concerns were mostly that some fish-safe products aren't axolotl safe, and there is no way to tell if the ingredients have changed from when it was known to be safe.
Bottom line: Unless money is no objection, if I were you, I would make sure that every other bit of equipment you need is purchases before thinking about buying a bacteria -containing product, and if you do buy one, go with a reputable product such as Seachem Prime.
Meanwhile, do you have the other equipment needed to clean your tank, like a siphon and something like a turkey baster to get chunks of poop out?
Given you have cats, you will need to keep a lid on the tank, but one of the ways to keep the water cool enough is to use "evaporative cooling", like, a fan blowing across the water. People in your situation usually get a mesh cover, so they can put a fan on top blowing down at the water while keeping cats out.
However, before going to get a mesh cover, find out what size your tank really is. I think you might need a larger tank, and in the long run, you would be better off getting it now. (Or if you have pet stores there that have periodic sales, getting it at the next sale.)