  We extend the asymptotic preserving and energy conserving time integrator for
charged-particle motion developed in [Ricketson & Chac\'on, JCP, 2020] to
include finite Larmor-radius (FLR) effects in the presence of electric-field
length-scales comparable to the particle gyro-radius (the gyro-kinetic limit).
We introduce two modifications to the earlier scheme. The first is the explicit
gyro-averaging of the electric field at the half time-step, along with an
analogous modification to the current deposition, which we show preserves total
energy conservation in implicit PIC schemes. The number of gyrophase samples is
chosen adaptively, ensuring proper averaging for large timesteps, and the
recovery of full-orbit dynamics in the small time-step limit. The second
modification is an alternating large and small time-step strategy that ensures
the particle trajectory samples gyrophases evenly. We show that this strategy
relaxes the time-step restrictions on the scheme, allowing even larger
speed-ups than previously achievable. We demonstrate the new method with
several single-particle motion tests in a variety of electromagnetic field
configurations featuring gyro-scale variation in the electric field. The
results demonstrate the advertised ability to capture FLR effects accurately
even when significantly stepping over the gyration time-scale.
