Is it fair to equate religious faith with the decision required by the hypothetical mountaineer? Certainly the common usage of the word faith is used in a variety of ways, and certainly some may say the mountaineer had faith he could make the jump. However, I would say the mountaineer's choice is better described as a calculated risk. She/he knows their capabilities, has extensive experience and therefore takes a risk. There is no faith that he/she will not fail, that the outcome is certain. On the contrary, it is a choice of best possible outcome without guarantee of success.It is different. Whereas the OP talks about a properly basic belief (God) as something of an axiom, James is contrasting faith with skepticism. This could be faith or skepticism of anything. Of course, what makes James interesting here is how his arguments may impact religious belief, but they don't necessarily have to.
James, being a pragmatist, wonders about cases where having an unfounded belief plays a genuine role in the outcome, and how we should treat these cases epistemologically speaking. He gives the example of a mountain climber who must make a leap across a chasm in order to survive. Supposing that this person is just barely able to make the jump to the extent that, if he believes he can make the jump, the boost on confidence and sure-of-footness will allow him to succeed. BUT, without the belief, he will fall just short. Is his belief that he can make the jump true or false? In either case (skepticism or faith) he will turn out to be correct. So, in that case, the better belief is the most useful one.
James does NOT endorse believing in falsehoods. If the climber could NOT make the jump, believing he could is bad.
James was responding chiefly to William Clifford who posited that it was unethical to believe in something without a proper degree of certainty. (ie. to Clifford, faith in God is unethical, and leads to unethical behavior/decision making). If anything, I think James does a good job of refuting Clifford. Though I'm uncertain how much work his arguments about faith really do. I'm a skeptic, and I always like to challenge my own positions, so James was really intriguing to me since his argument isn't super weak. Theists are pretty good at bringing the weaksauce (like this reformed epistemology crap). James makes a decent case.
If you aren't familiar with Clifford, I linked it below. If anything, his opening parable (paragraph 1) and Clifford's interpretation (paragraph 2) is must read material for all atheist's and skeptics. You can skip the rest as it just goes into more detail from there.
https://people.brandeis.edu/~teuber/Clifford_ethics.pdf
Contrast this with religious faith, which to my mind represent a strong belief without sufficient evidence to support, or really belief in spite of conflicting evidence against the belief.
To me, these are not the same uses of the word faith.