Ignoring the various tangents and side alleys this topic has taken.
Is there a moderate Islam? Yes, there is. It is what is practiced by moderate Muslims, who do exist, believe it or not. Is it an organized branch of Islam? Not that I know of.
Is there a moderate Christianity? Yes, there is. It is what is practiced by moderate Christians, who do exist, believe it or not. Is it an organized branch of Christianity? Yes, but only because of the influence of humanism and secularism.
Taken on their own, both Islam and Christianity are extremely intolerant of any other religions, even to the point of motivating physical violence and even genocide against those other religions. Both religions have a set of very draconian laws and punishments for all kinds of infractions, even non-existent ones invented by the religion itself. Both religions also have many peaceful and socially and personally beneficial elements in addition to the malice and mayhem.
Moderate Muslims and Christians both are able to be moderate by not taking their religions so seriously. They are able to overlook or downplay the malicious negative elements and concentrate on the beneficial positive elements.
Similarly, the Muslims and Christians who really take their religions seriously are not allowed to overlook the malicious negative elements and even find that they must embrace those elements. That is when either religion becomes a very definite problem.
What allows Muslims and Christians to be moderate is a society that does not allow that religion any political control. A secular government, preferably based on humanistic principles such as those that the US Constitution and Bill of Rights were based on, is an ideal environment for Muslim and Christian moderates, though one must still guard against religious fanatics exerting too much influence.
Obviously, Islamic governments which exercise Sharia law are not secular governments that are needed for moderate Muslims to thrive. Moderates will still exist in such societies, but they must remain closeted in order to survive.
Similarly, any government that would implement "God's Absolute and Immutable Law", AKA "Biblical Law", would not be a secular government and as such would drive moderate Christians deep underground.
Islamic governments, Christian governments. No difference. Except that secular governments have existed and operated in the "Christian" West for far too long for the governed to readily hand control over to the implementation of "true Christianity's" "God's Law", whereas Sharia Law has a much longer history in Islamic societies.
Of course and unfortunately, even when you do have a secular government that promotes religious tolerance, you will have members of that government's society who take their religions far too seriously and who ache to impose all the malicious negative aspects of their religions on the rest of society, including the draconian imposition of their own particular "God's Absolute and Immutable Law".
Muslim extremists, Christian extremists. No difference. Except the West has far more experience keeping the Christian extremists in check.