Why are there negative energy particles inside a black hole?
As far as I know, any observer (inertial or otherwise) with 4-velocity $U^\mu$ will measure the energy of a particle with 4-momentum $p^\mu$ to be $U^\mu p_\mu$. Since no observers have spacelike 4-velocities, no observer would measure the energy of a particle to be $K^\mu p_\mu$ inside the black hole. Trying to define the energy 'as measured at infinity' strikes me as dangerous: we can only measure local quantities in GR, so the only way to talk about an observer measuring something far away is to imagine some signal travelling between events. But this cannot happen if our particle is behind the horizon.
What I believe Hawking is trying to say is that it’s OK for the quantity $K^\mu p_\mu$ to be negative, precisely because it isn't energy. There is a brief moment, as the particle approaches the horizon, during which the particle has negative energy, which would be problematic were it not for the brevity of the moment (forgive the hand-waving). But once inside the black hole, this quantity doesn't correspond to energy, and so we no longer have a problem. Indeed, a typical observer inside the black hole would have 4-velocity something like $-\partial/\partial r$, and so would measure the particle's energy as $-p_r$, which is positive, since $p^r$ is negative.
Note: the quantity $K^\mu p_\mu$ for any Killing vector $K^\mu$ is conserved along the particle's geodesic.
I assume the mostly minus metric convention.