I have had my ladies for 1 year now never cycled a tank or used a filter do a 90% water change on them weekly. I use a sponge filter just to get the extra missed stuff ill leave a link. As for the babies I have them in a small tub half way filled may bet a few inches over their heads and I do water changes every 2 days they have a small betta filter in with them just to be safe
I don't put declorinator as it dissipate s after time anyways and axolotyl breath air not water so chlorine is fine as long as you have safe tap water to drink its ok for them. As long as you keep the tank clean weekly and take out the 90% ammonia from them toileting they will be fine as you see they are in the tubs with no filter.. They are very easy to manage. Clean that tank with water and a safe sponge I'm sure that algee is not good. I have 6 axolotyl currently two large 1 medium 1 small and two babies 3 inches also I have a bundle baby blue gills hatchlings I'm raising now same method. So conclusion
1.Ditch the filter for a sponge filter they hate currents from that hangover filter.. but mines love the bubbles the sponge filter makes.
2. Do 90% water changes for first few months weekly even with sponge filter and 50% every other week after sponge has cycled. Although you don't need a filter at all if you stay the 90% method. I do both just to keep them disease/fungus free
3. Chlorine won't hurt them it goes away when contact with air in 24 48 hours and it could help kill any fungi in tank. All other products may harm axolotyl and they prefer hard water anyways.
4. Remember they were found in a lake I'm sure the babies will be fine with tub half full. Those lakes are deep in ther natural habitat ..Mexico so I think if you want to be safe...Put Babies in tub few inches over there heads are fine. As long as its enough for them to swim/exercise and come up for air without exhausting themselves so I would say put them in a tub with about 5-6 inches over there heads..they will have room to stretch out at least
You can feed the babies red wigglerrs chopped up the length of their head from mouth to back of head so not to big to swallow with sucking back in. It will go straight in the first time. Blood worms are so messy and foul the water and the pellets suck you have to get them soft they cloud the water etc. Worms keeps the water clean and you know how much you feed them
https://www.amazon.com/gp/aw/s/ref=is_s?k=corner+bio+filter+20+gallon
Filter for my 40 gallon breeder I use ...
Filter for babies in tub I use
https://m.ebay.com/itm/Elive-Betta-Filter-w-Glow-Plant-Aquarium-Filters/182624253116