It's just that right now, even if I do specify exactly how I think the generating process works, I still need to work out how to do the estimation.

If you can implement a good simulation of the generating process, then you are already done - estimating is as simple as ABC. (Aside from the hilariously high computing demands of the naive/exact ABC, I've been pleased & impressed just how dang easy it is to use ABC. Complicated interval-censored data? No problem. Even more complicated mixture distribution / multilevel problem where data flips from garbage to highly accurate? Ne pas!)