  An opto--electro--mechanical system formed by a nanomembrane capacitively
coupled to an LC resonator and to an optical interferometer has been recently
employed for the high--sensitive optical readout of radio frequency (RF)
signals [T. Bagci, \emph{et~al.}, Nature {\bf 507}, 81 (2013)]. Here we propose
and experimentally demonstrate how the bandwidth of such kind of transducer can
be increased by controlling the interference between two--electromechanical
interaction pathways of a two--mode mechanical system. With a
proof--of--principle device \new{operating at room temperature, we achieve a
sensitivity of 300 nV/Hz^(1/2) over a bandwidth of 15 kHz in the presence of
radiofrequency noise, and an optimal shot-noise limited sensitivity of 10
nV/Hz^(1/2) over a bandwidth of 5 kHz. We discuss strategies for improving the
performance of the device, showing that, for the same given sensitivity, a
mechanical multi--mode transducer can achieve a bandwidth} significantly larger
than that of a single-mode one.
