  We discuss plans for the E302 instability-efficiency experiment, starting in
2024 at the recently upgraded FACET-II facility at SLAC National Accelerator
Laboratory. The beam-breakup instability will be the main area of study for the
E302 experiment. With the imaging spectrometer at FACET-II, we introduce a
novel technique for observing and quantifying the amplitude growth of the
trailing bunch due to the transverse instability. Using the transverse position
on the spectrometer screen and the transfer matrix of the magnetic lattice used
for the spectrometer, we aim to extract a $x'$-$E$ charge distribution that can
be used to quantify the amplitude of the beam. By varying the trailing bunch's
charge and, hence, the beam loading of the accelerating field, we aim to adjust
the wake-to-beam power transfer efficiency in the E302 experiment. We plan to
quantify the amplitude for different configurations of the beam charge and,
hence, investigate the relationship between the beam-breakup instability and
efficiency. We use a combination of particle-in-cell (PIC) codes to simulate a
beam-driven plasma wakefield accelerator from start-to-end with a FACET-II-like
spectrometer and demonstrate the methodology that will be used for the
instability studies at the E302 experiment.
