Correct. That makes the problem much more difficult.
A further difficulty is that we cannot replace something with nothing. Christians, on the whole, have lost confidence in their beliefs. Mohammedanism cannot be replaced with mindless mush like "democracy". Democracy is just a way of picking rulers. Set up a democracy in a land of mohammedans, and they'll pick mohammedan totalitarians as rulers; they'll impose sharia as the law of the land. If we wish for them to pick better rulers, they have to not be mohammedans. We Christians need the confidence to say "MOHAMMEDANISM IS WRONG. CHRISTIANITY IS RIGHT. JESUS CHRIST IS LORD." Until then ... SSDD.
Forget targeted measured response. We take a piece of a city and a week later the bad guys have it back. Before we walk away we need to leave them with nothing more than smoking ash and weeping widows, to the extent we are nice enough to leave many widows. We didn’t worry much about that at Dreasden or Hiroshima.
Too nasty for your taste? Bring all our people home. Now. Sure we will see terrorist arracks from the region. Read the papers lately? We see them now.
In the future, trace any attack to its source and remove from the Earth any trace of that place. Probably want to avoid nukes, thoigh they would be cheaper, but we are very good at killing people and breaking things if the policy people take the cuffs off.