Here's one using a flanger to add a variable delay to each pitch voice. Maybe this sounds better than the last clip. Technically this adds some pitch modulation but it's slow enough to not really matter. The pitch voices go through flanger as L/R with nonzero LFO phase, so the timing can vary in different ways. A more random version could be to have separate LFOs from two delay/flange blocks (or use both LFOs in one delay block) at different rates, maybe running each LFO a bit faster when that signal is muted.
https://dl.dropboxusercontent.com/s/3sgjjesrwxw9a66/doubler-5-21-20.mp3
I can think of some better ways to control randomization, but the Axe-FX doesn't really have the tools for it. For example, switch the delay/detune/etc. values immediately when a picked note is detected, instead of using random LFO which takes 33 ms. Only jump between nearby values, or restrict the amount of each jump or probability of larger ones somehow. Swap channels only when time offset is at minimum, but also have it go into a state where this is guaranteed (first stepping down to min. time) if that hasn't occurred in some time.