So why does God want Pharaoh to do the sending?
The simple answer as far as I can tell is because Pharaoh did the enslaving.
If we take the model metaphor of God as parent and humans His children, it would go something like this:
The parent sees his his one kid beat up on and tease his other kid. The parent can pick up the taunted kid and remove him from the torture directly. But if he does that, the bully kid doesn't really learn anything and, not only that, the tortured kid learns that bullies don't necessarily suffer any consequences. So even if the parent thinks the bully kid will not really change his ways if he is punished, he might still punish him in order to teach the other kid that ill behavior does have consequences.
So it is with Pharaoh, the bulliest of bully kids.
Now, though, why should God reward him for sending the Children of Israel out of Egypt?
This is a bit trickier but I will offer at least one suggestion--one for tonight, anyway. Maybe another one for tomorrow night.
Jewish thought includes a notion of reward as well as punishment. One is to be rewarded for fulfilling God's will just as one is to be punished for violating that will.
Pharaoh was commanded, that is, he was given a mitzvah to send the C of I out of Egypt. Ultimately, he fulfills that mitzvah so he should get a reward for it, right?
Maybe you cringe a bit when you read that. After all, Pharaoh really tortured the C of I and, in any event, only released them after his kingdom was a shambles and his own first born died. Nonetheless, I would still say he is deserving of reward of some kind.
Note that the reward doesn't go to Pharaoh directly but rather to the Egyptian people as a whole (see yesterday's posting). This is true for the other examples brought by the M'chilta as well.
But also remember the parent/child metaphor. If the parent forces the bully child to 'make nice' the parent will want to reinforce that behavior by rewarding the child. You might argue that, again, Pharaoh is not really going to change his ways no matter what God does. But I would then add that, again, another reason for the parent to reward the bully when he finally 'does the right thing' is to show the other child that the parent will reward good behavior.
You might ask, though, if Pharaoh really fulfilled the mitzvah at all since, after all, it was done under duress. Well, we'll look at that tomorrow.