>"I'm aware that stirring up the soup and mixing in natural material will help the waste break down quicker but I was wondering if adding a pinch of septic tank pro biotics would help speed the process along."
This comes rather close to my day job - the clean up of contaminated waste sites. Various practitioners and vendors offer their propriety bacteria that are supposedly optimized for different contaminants. The thing is, at least a dozen species of soil bacteria can metabolize gasoline and diesel-range hydrocarbons, amazing though that may seem. What you deposit is even easier for them to digest and is already loaded with bacteria.
If you want to help the process, here are things that universally help aerobic bacterial action: atmospheric air (poop in loose soil, not tight clay), water (pee in the same hole), fertilizer (pee in the same hole), and warmth (a south-facing, exposed-to-the-sun patch of soil will be >10C warmer in the summer at a 6" depth than a shaded area).
I don't suggest the following for several reasons, but if you (1) added a pinch of Scott lawn fertilizer, (2) stirred well, and (3) dusted the replaced surface with carbon-black soot (to increase solar heating); you'd increase the degradation rate at least 10-fold. Sewer treatment plants, with active management of oxygen and nutrients, can degrade human waste in about a day. In more controlled settings, I can take toxic-waste-containing water to drinking water standards in less than an hour.
An afterthought: for how helpful peeing in the hole is (especially in the dry summer Sierra), and to avoid peeing on the shorts around your ankles, how about this: Dig the hole. Pee on the spoils pile. Do #2 in the cat hole. Stir in the wettest soils. Finish filling hole. Cover with leaf litter.