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If we "remove religion" from our understanding of this conflict, will that matter to the other who has declared war on us and shall NOT "remove religion" from HIS understanding of this conflict?

Once again, one of the radical Islamic groups primary selling points for recruitment is the idea – propagated by them and reinforced by some Christians – that the Christians are waging a New Crusade against Islam. They have only to link to some of the Free Republic posts to convince potential members that the Crusade is a fact.
Remove that incentive to recruitment.

Should the priority be to advance Christianity by the elimination of competition, of the protection of We, the People?

I think you missed the point.
No matter what we do, they will hate us.
No matter what we say and/or actually believe, they will still play the Crusade Card.
Our "removal" of religion would have exactly ZERO net effect on their recruitment policies and those policies' efficacy.
One of the primary reasons for this is that it is so easy to believe, because Islam IS at WAR with Christianity.
Projection is rife in the Muslim worldview. Because they are in perpetual jihad against all others, they will have a natural tendency to believe that all others are waging religious holy war against them.