Firstly I would advise against calling those who oppose the Muhammad pictures "extremists" or "fanatics". Although some of those users might have violent tendencies, protesting against what you feel is an insult to your faith is hardly extremist behaviour.If it were so you would have to call a large number of Christian leaders fanatics for opposing the "Da Vinci Code". Infact by past standards the response of the Muslim groups to this issue has been exceedinly passive, not even a fatwa has been issued yet.
That being said, to compromise on principles due to pressure from certain groups is hardly desirable. However experience shows that Muslims are a very difficult group to negotiate with. The fact that the concerned Wikipedia article has 3 talk pages to explain the issue to offended Muslims is proof of this. This tendency (of not budging from their fundamentalist religious views) among those of the Islamic faith is described by some as "fanaticism" and by others as "admirable devotion", and it is this trait which perhaps sets Islam apart from other major religions.
The point I a trying to make is that you cannot and will not succeed in convincing Muslims to accept that image, however hard you try. In fact you would have much more chance of convincing non-Muslims of why the image had to be taken down. Wikipedia seems tobe trying to make a point. One wonders what it is?