Is there a way to pause cycle layers without having to activate manually again?

One other trick, while I'm thinking about it. There's a "Red Monster Live" puppet at a desk among Okay Samurai's puppets that had some occasional background animations. I think what they did was to nest the cycle you want to play occasionally in another cycle. The nested cycle works as you have it now as a single frame in the outer cycle, but the outer cycle has a high "Advance Every" parameter so it runs much less often but only needs to have a couple of layers to slow down how often the inner one runs. So that could be an option as well. (if you download the puppet and open it, search for "starcycle" in the search box at the top of the puppet layers panel, it looks like it was a very similar background feature)

I've seen some really clever puppets that use nested cycles to do all kinds of crazy stuff.

I haven't tried this, but I think maybe you could tag the whole cycle as a "Blink" and use the Auto blink behavior with a low "Blinks per Minute" frequency to semi-randomly trigger it periodically?

I have not tried it, but in the Cycle Layers behavior (in rigging mode) there is an option to pause layers. It is mentioned in this thread. Is it possible to pause a cycle by, say, holding t... - Adobe Support Community - 8814448

You can select the layer you want to pause and add a "Cycle Pause" tag to it. Beyond that, I am not sure.