calculate force on object between capacitor plates without knowledge of area
In your current thought process, it looks like you're trying to find the charge on the capacitor (You denote it generically as \$Q\$, I'll denote it \$Q_{\text{cap}}\$), which is proportional to the area of the plates. However, since we know that \$Q_{\text{cap}}/C = V\$, you'll divide by the capacitance (which is itself proportional to the area), and your final result is area-independent.
A more straightforward formulation uses:
$$V = -\int \vec{E} \cdot d\vec{l}$$
Assuming the dielectic is uniform throughout the capacitor and edge effects are negligible (i.e. the particle is far from the edges), this turns into simple multiplication/division to relate the field to the capacitor voltage. You can subsequently relate the electric field and \$Q_{\text{particle}}\$ to the electrostatic force on the particle that should hover:
$$\vec{F}_{\text{coul}} = Q_\text{particle}\vec{E}$$
and of course this should be equal in magnitude and opposite to the gravitational force on the particle.