Distinct fractions with parsimony on denominators looks like the issue of the average order of the Euler totient function $\phi (n)$. Which is well known, in that it means the exponent of n you need is easily at most $1 +\epsilon$, and in fact the real average order is more like n/loglog n .
The condition on the numerators is slightly troublesome, though. Just taking the p/q where q is fixed and p coprime to q is going to have the sum of p's around half the sums of q's, on average (or a bit less ...). I think this can be worked round.