  At low temperatures, microwave cavities are often preferred for the readout
and control of a variety of systems. In this paper, we present design and
measurements on an optomechanical device based on a 3-dimensional rectangular
waveguide cavity. We show that by suitably modifying the electromagnetic field
corresponding to the fundamental mode of the cavity, the equivalent circuit
capacitance can be reduced to 29 fF. By coupling a mechanical resonator to the
modified electromagnetic mode of the cavity, we achieved a capacitance
participation ratio of 43 $\%$. We demonstrate an optomechanical cooperativity,
$C$$\sim$40, characterized by performing measurements in the
optomechanically-induced absorption (OMIA) limit. In addition, due to a
low-impedance environment between the two-halves of the cavity, our design has
the flexibility of incorporating a DC bias across the mechanical resonator,
often a desired feature in tunable optomechanical devices.
