Given those two options if fight we must, fight with conviction and determination.
One problem is that you are presenting the entire situation as either we fight against the Islamic world, or we stop opposing them and instead attempt to promote internal reform.
Why can't we, instead of dogmatically clinging onto one doctorine or the other, instead choose the action that will be maximise our ideals in a given situation.
For instance, perhaps promoting economic prosperity in the Islamic world in one instance is best (as improving material conditions will likely also result in social development, especially if the Islamic world can be tied to the international capitalist system), whereas in another situation using the military to defend our interests is best.
If we just choose one or the other, because such an approach fails to account for the complexity of the situation (the complexity of religion, of the social dynamics within the various nations, of humans in general) then of course it will be doomed to failure: If we simply accept fundamentalist Islam, obviously our greater liberal and social ideals will not be promoted. If we completely ignore the "arab street" and instead attempt to bomb our way to victory, obviously we will only increase the gap between Islam and liberalism and radicalise increasing sections of the population.
Neither approach is going to work solely on its own.