You seem to have an interesting definition of fundamentalism, one that I don't think the general population holds. As far as I can tell, when most Americans use the words "fundamentalist Christian," they are referring to Christians who hold to the orthodox doctrines of Scripture. And to pluralistic American society, the most offensive and noteworthy of all orthodox doctrines (and thus the one by which they measure a person's fundamentalism) is the doctrine of salvation solely by faith in Christ, apart from other religions. In other words, fundamentalists believe that all non-Christians are going to hell. And if this is our working definition, then I am most certainly a fundamentalist, since I believe that all people who practice non-Christian religions (theistic or atheistic) will go to hell. If we can all be honest for a moment, I think it's obvious that this doctrine is ultimately the reason that many people perjoratively label many Christians as fundamentalists. But since it is such an important doctrine, I feel it important to note that I fully embrace it.
But do I reject logic? Certainly not. And if this is how you define fundamentalism, then I am not a fundamentalist. To be sure, I do not believe that God is a slave to logic or anything else. However, to even suggest such a thing is to make a false dichotomy between God and logic. It can be clearly inferred from the Scriptures that logic is one of God's attributes. God does not act contrary to his own attributes, and so God does not "transcend" logic anymore than he transcends holiness or righteousness. All of these things are a part of who God is. None of the doctrines of the Bible are illogical. Neither our belief in the existence of God, nor his supernatural characteristics, nor the resurrection of Christ and salvation by faith in his Gospel, are illogical beliefs. I do not have faith in the existence of God or the resurrection of Christ; rather I take these as objective truths. Biblically speaking, faith is a
hope in a currently unrealized promise of God. As it says,
Now faith is the assurance of things hoped for, the conviction of things not seen. (Hebrews 11:1)
Therefore my faith is in God's promise that by trusting in Jesus Christ, I can be saved from my sins. The existence of God is an objective fact that can be verified by the Bible's historical record of his dealings with man. The same is true of the resurrection of Jesus. Often times people state that the existence of God is an illogical proposition that must be taken on faith. This is flawed because it demeans faith by portraying it as illogical. Furthermore it ignores the fact that God's existence is an objective reality. Even God's enemies believe in him, as it says,
You believe that God is one; you do well. Even the demons believe--and shudder! (James 2:19)
Do demons have faith in God? Certainly not. They disbelieve his promise that he will destroy them for their rebellion. And yet they believe in his existence. So clearly the existence of God is not something that has to be taken on faith. I think that all too many times, non-Christians object to our portrayal of the resurrection of Jesus as a definitive truth, stating rather that it is "a matter of faith." Perhaps non-Christians think that God will not fault them for not having faith, i.e. they think that they can reject Jesus and not go to hell for it. Alas, this is speculation on my part, as I can't know the ultimate motives of non-Christians.
But to address your initial point, it seems that your brother may be misunderstanding certain Christian doctrines if he rejects logic. One ought not to believe in the doctrines of the faith and reject logic. Logic is rather instrumental to understanding the Scriptures, so any Christian who rejects it is experiencing a cognative dissonance of sorts.