you are in effect cycling your tank with your axolotl in it, the cloudiness is bacteria bloom because there isn't a large enough colony established in the filter.
your best course of action would be to remove and tub your axolotl until the cycling is complete. when the bacteria colony is removed from the filter it has to undergo cycling again to rebuild the bacteria colony.
chemical filtration.. activated carbon, replaced every month.. zeolite, re-energised by washing in salt water.
mechanical filtration.. sponges, squeezed out/rinsed out in old tank water.
bio-media.. ceramics/plastic balls etc. rinsed/washed through using old tank water.
always useful to have bottle bacteria to top up if needed.