Generating all the permutations can be done easily using constraint programming. Here is a short model using MiniZinc for the above example (note that we assume that x will contain n different values here):
include "globals.mzn";
int: k = 2;
int: n = 4;
array[1..n] of int: x = [0, 1, 2, 3];
array[1..n] of var int: y;
constraint forall(i in 1..n) (
y[i] in {x[i + offset] | offset in -min(k, i-1)..min(k, n-i)}
);
constraint all_different(y);
solve :: int_search(y, input_order, indomain_min, complete)
satisfy;
output [show(y)];
In most cases, constraint programming systems have the possibility to use a random search. However, this would not give you a uniform distribution of the results. Using CP will however generate all valid permutations more efficiently than the naive method (generate and test for validity).
If you need to generate a random permutation of your kind efficiently, I think that it would be possible to modify the standard Fisher-Yates shuffle to handle it directly. The standard algorithm uses the rest of the array to choose the next value from, and chooses the value with a probability distribution that is uniform. It should be possible to keep a list of only the currently valid choices, and to change the probability distribution of the values to match the desired output.