Dubai is part of the UAE; it doesn't count as two governments. Besides, unrepresentative tyrannies that follow Islamic law are hardly reliable and given their history the word "ally" is not appropriate unless followed by "of convenience". By treating those governments as legitimate (which they are not) we are legitimizing despotism. Since they are not representative governments, and rule by force anyway, how much do their domestic politics really affect their policies? Not a whole lot, not in the way that domestic politics affect policies in the West.
I would very much like to see this list of "Islamic" governments that are our allies. I'm going to take a wild guess and say that list can't be particularly long and consists entirely of countries whose continued existence depends on the presence of US military forces.
France's government is legitimate because it is representative. However, they are not an ally in any meaningful sense and have not been in quite some time (since 1968 at least). Plus they were in up to their eyeballs with Saddam and never supported the mission, even if no contribution were expected of them. So them not having troops in Iraq is hardly unexpected.
I am rather shocked to see someone seriously propose that the primary front in this war is in law enforcement. That is precisely the approach taken in response to the 1993 World Trade Center bombing. Didn't 9/11 wake you up to the inadequacy of that approach?
Moreover, the only convincing of public opinion we need to do in the Arab world is to make them fear us more terribly than anything else. The "Arab street" is a joke - they will do what their heavy-handed tyrants force them to do. With the tyrants you insist must be kept in place, no possible influencing of public opinion can occur. In fact, it is quite arguable that keeping the Emirs, Sauds, and other despots in power is precisely what sets opinion against us - but for the US' support, these tyrants might not rule them by force.
This is a military war, we've already fought two significant campaigns in it. To say otherwise is to twist the very meaning of the words into meaninglessness.
In fact, our military presence in Iraq will be modified due to domestic public opinion.