I would instead look at the various steps in the filter, and generalize the parameters of those steps, which then generate universes with various levels of noise / age at first space-colonizing civilization. If you have fat-tailed priors on those parameters, I think you'll get that it's more likely for there to be one dominant factor in the filter. Maybe I should take the effort to formalize that argument.

Another way of thinking about the filter/steps is as a continuous developmental trajectory. We have a reasonable good idea of one sample trajectory - the history of our solar system - and we want to determine if this particular civilization-bearing subspace we are in is like the main sequence or more like a tightrope.

If the development stages have lots of conjuctive/multiplicative dependencies (for example: early life requires a terrestrial planet in the habitable zone with the right settings for various parameters), then a lognormal distribution might be a good fit. This seems reasonable, and the lognormal of course is extremely heavy tailed.

On the other hand, one problem with this is that seeing a single trajectory example doesn't give one much evidence for any disjunctive/additive components in the distribution. These would be any independent alternate developmental pathways which could bypass the specific developmental chokepoints we see in our single example history.