The Combinatorica package was available with earlier versions of Mathematica. You might see if it is included in your version. That included a NextPermutation function which would generate each subsequent permutation, one at a time, instead of creating all the permutations at once. If that works then you might trade longer execution time for less memory usage.

If you got NextPermutation working then you could generate and test each permutation and only save those which satisfied your conditions. I once found the source to the Combinatorica package on the net and was able to extract that single function for a problem I was working on. If you try this then please test it very carefully to make certain that it is correct for your problem.

Different idea, do not know whether this will help or not, probably not as much. Is

5 <= Total@#[[11 ;; 15]] <= 5

equivalent to

Total@#[[11 ;; 15]]==5

or perhaps even

#[[11 ;; 15]]=={1,1,1,1,1}

Another idea. Could you generate the permutations not including those 5 1's in columns 11-15 which might make for fewer permutations to generate and test. Perhaps something like

base = Select[Permutations[{0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0}],5<=Total@#[[1 ;; 10]]<=7&]

and adjust your code to understand that columns 11-15 are not present or you insert those 5 1's into the middle of each permutation after you generate it if needed.