Factoring into primes doesn't really affect the lower bound on complexity.
Note that for N=2^64, the number N! is about 10^(10^20.5), i.e. just the number of digits in N! is itself about 3×10^20. The number of bits in N! is itself over 10^89. Every prime number from 1 to 2^64 (over 10^17 primes) is a prime factor of this number N!; if an algorithm is supposed to work with those many primes, it's clearly no longer O(K) as K could be small relative to N.
An algorithm (like one of the proposed encryption schemes) that produces, say, one of 2^1024 permutations may be “good enough for practical purposes” (as 2^1024 is much larger than the number of atoms in the universe), yet as a fraction of the space of all possible permutations, it is dizzyingly insignificant.
There are ways to do it with greater coverage of the space between 4 and 2^64, but I would defer to better judgement before doing so.