Well, you’d have a very hard time producing neutral hadrons, and nearly as tough a time producing neutrinos, unless your energies were up near the mass of the Z. But yeah, it’s possible.
I wouldn’t say “it’s possible” but rather “it’s exactly the thing that happens under OP’s conditions”. Neutrinos require higher energies than you’d find around your house, but so does everything else outside of photons, so as long as you allow accelerating the particles (per the OP), neutrinos are in the game as much as the rest. And as long as you’re producing hadrons, neutral hadrons are in the game as much as charged hadrons.