Exodus Rabbah 1:26 brings this midrash: one day when Moshe was a child he grabbed Paro's crown and the court magicians counselled Paro to have him killed lest he usurp the throne. Ultimately a test was proposed and Moshe passed (with Gavriel's help), so he was allowed to live. But it's not unreasonable to think that the magicians would continue to caution Paro about this, and he might have worried about it even without their encouragement.
So, if he's already worried about his adopted grandson challenging him some day, and then he hears that he's attacked one of Paro's officials (albeit a very minor one), he might well decide that this challenge is dangerous and must be stopped, no matter how much he enjoyed playing with him as an infant.
Finally, as SethJ noted in a comment, this is not unusual for rulers.