Super Mega Pet will only fall asleep if he feels tired. When you see the Z's coming off him, then click sleep to give him some rest. This is random, he sleeps at anytime and for a random length of time. When he does sleep, it is usually only very briefly.
While sleeping, he won't eat no matter how hungry he is. It is a good idea to leave him alone until he wakes up. Although the digital tank can still be cleaned and toys added.