Yep, I've learned when you plant your lily tuber, you put granular fertilizer (the bigger the middle number, the better ... i.e. 10-35-10 is good) in the bottom before the kitty litter. Then soak the litter so it makes a "mud", then put the tuber into the litter. Later, when you want to fertilize it, you can use the plant tabs made for ponds, but they are expensive. I've also learned that using tomato spikes are good (they are the high middle number again) or using the granular can be used by wrapping it in a paper towel, pushing it down into the kitty litter. The paper towel keeps it intact until it's in the litter, then slowly decomposes to allow the fertilizer to slowly release. Good time to fertilize after you pot them is after they are growing well, and water is warm, to promote more blooms.
If you get fertilizer into the water, that can cause algae blooms, so want to try to keep it contained in the litter, thus the spikes or paper towel wrapped granular. If you watch at the end of the season, you may find the lily fertilizer tablets on sale, if so, grab them up! They are small, easy to push into the pots during the summer, and proper numbers. Also, check them out when your stores have them (lily fertilizer tablets) to see then numbers they have, then look for similar numbers on fertilizer that is much cheaper. I bought my granular at the end of the season cheap, so I have a good supply for this year.
The kitty litter pan will work just fine. They say tuber type lilies and lotus like to grow "around", so that's why they recommend a round pan for them to be in. I would guess the litter pan would be rounded just fine for them to keep growing around. The problem with a square pot is that if the plant roots gets to a corner, it's as likely to go "up" rather than make the hard turn and keep growing around, and once the roots get out of the pot, they go nuts! Believe me, you want the tubers in a good solid growing pot. If it's flimsy, they will "break out" and cause havoc in your pond. The problem is that the roots will grow until they find another pot, and then root in there, and once they are rooted, it's very difficult to pull the plants apart if you want to move a pot. I've had all of the above happen, and as I always say, learning as I go.