I do know that any such oscillators must operate somewhat like a wave pool, with an advancing wall of live cells. When that wall dies, there must be another wall behind it, and there cannot be any holes in the wall, or safe spaces to hide if the next wall has not yet fully formed.
I haven't yet found any examples (even when looking through jslife, Sokwe's jslife-osc-supplement, and the Oscillator Discoveries thread), but I have found a few near misses that should help illustrate my point:
x = 61, y = 11, rule = LifeHistory
The wall in the first oscillator doesn't go far enough out before reforming, so no travel in either direction (center to sides or vice versa) is possible. In the second oscillator, the wall is broken in one phase, and the agent can pass right through that gap. In the third oscillator, the wall remains intact, but the agent can still get through from top to bottom by waiting at the location of the red cell, and then continuing once the wall has passed.
As for what periods to search, P3 through P6 seems like the sweet spot, with an emphasis on P4. However, another, higher-period option is possible using signal terminations, which are stable at rest; thus, only the times around the signals' arrivals are crucial, as the stable walls do the rest of the work.