“Fundamentalist” has a particular connotation, especially in the US of (1). The Bible is literally true, including a supernatural Creation and Flood. (2). Christianity is the only true religion, and all other religions and secular philosophies are misguided or evil. (3). The ultimate society would then be a Christian theocracy; at a minimum, any attempt by secular society to limit or interfere with Christian values should be defeated.
Now based on his writings, Lewis did not agree with most of the above. He did not think it necessary to postulate that the first humans with souls were created supernaturally. He regarded salvation as personal, individual, and uncoercible, and not as the result of any worldly institution. He spent his career addressing the fact that intelligent sincere people could doubt Christianity. I don’t know if Lewis ever had knowledge of or wrote about American fundamentalism; his main concern was a British society already much more athiest than anything imaginable on this side of the Atlantic.
But Lewis believed that Christianity was literally true. That it was the final, indeed the ONLY answer to the human condition. He believed that only the determined actions of good people (ultimately, Christians) kept human society from collapsing into unchecked evil. He even went so far as to say that the problem with medieval society wasn’t that it was too Christian, but that it was not (truly) Christian enough. Without insisting on forcing anything on anyone, Lewis was uncompromisingly Christian. So what can you call Lewis’s stance without making it sound like he was a Bible-thumping backwoods preacher who denounces “Evilution” at revival meetings?