Why one uses 了 and the other one doesn’t?
Overall, your analysis is correct. It's indeed for the sake of “even amount of characters (4 - 4)” in this case. It might be more obvious if it writes: "吃饱就睡，醒了就吃" or "吃饱就睡，睡醒就吃".

吃饱了就睡 is also correct.