So, the densest a planet can reasonably get is effectively making it out of solid iron. (Making it out of denser materials is unlikely given the relative rarity of those materials, and a big chunk of iron could be something like a piece of a stellar remnant after some star-shattering cataclysm.) The moon has a volume of approximately 21.9 billion cubic km. Iron has a density of 7.3 billion tonnes per cubic km. A moon-sized hunk of iron, then, would mass approximately 1.6 * 10^23 kg. That's another order of magnitude denser than it currently is, but still only about a tenth of Earth's mass (rather than its current figure, 1%).
So, can we make Earth lighter? The least dense planetary masses in the solar system are the gas giants, but they're also the most massive. Happily, the good people at Harvard have a lifeline for us. Turns out, if you're a good distance away from the star, the minimum core mass for a gas giant is circa 0.2 earth masses. The authors in fact postulate that Uranus and Neptune were originally formed as much smaller planets.
If we assume (not wholly accurate, but we're working with Fermi numbers here) that the core of such a planet is composed entirely of ice, and that the overall density of the planet averages 0.9 g/cm^3 (heavier than Saturn, but a much lower gas-to-core ratio), we still can only manage to get an earth-volume planet down to around 9 * 10^23 kg. This definitely would result in a more equal gravitational relationship (rather than the 100-to-1 relationship with our moon), but the disparity in diameters is so tremendous that there's no explicable natural way (without dipping into neutron stars or black holes, which you said you'd prefer not to do) for the larger to orbit the smaller.
Geometry does tell us that you'd only need to increase the radius of your super-dense moon fivefold to give it a hundredfold increase in volume and mass (at which point the planet/moon relationship could be established), but it would still be five times bigger than the moon.