Just a partial answer, but the proposed inequality doesn't hold.
Take $p = [0.2, 0.8], q = [0.1, 0.9]$.
Then $H(p) = 0.2 \log(5) + 0.8 \log(1/0.8) \approx 0.5$,
$H(q) = 0.1 \log(10) + 0.9 \log(1/0.9) \approx 0.33$
and $KL(p, q) = 0.2 \log(2) + 0.8 \log(0.8/0.9) \approx 0.04$.