The answer to your question is yes. This can be easily deduced from results in a recent paper written by Sara Billey, Chris Burdzy, and myself, arXiv:1209.0693. Clearly the number of permutations you are considering is less than the number, a_n, which have interior peaks (what you are calling interior local maxima) at only odd indices and valleys (what you are calling interior local minima) anywhere. A simple computation using the Theorem 1.1 in the cited paper shows that a_n is at most p(n) 2^{2n} for some polynomial p(n). But since (2n+1)! grows faster than [(2n+1)/e]^{2n+1}, the ratio clearly goes to zero.
Cheers, Bruce Sagan
PS Thanks to Richard Stanley for pointing out this post to us.