I am only sort-of-well-read about the portobello and white button mushrooms. For those types, do not inoculate the compost right away! A few thoughts for now:
1. The mushrooms actually live off of the bacteria bodies that did the composting. Composting produces ammonia that is toxic to the mushroom growth. The ammonia must be broken down by more bacteria before the mushroom body can grow. If there is any ammonia smell don't inoculate.
2. There are a lot of special steps given in the huge book I am reading, but they are primarily geared to bigger growers. The key point of all the steps is to make sure the mushrooms can out compete bacteria, molds and other fungi.
3. Pasteurizing the compost seems reasonable, and I will try to do it. I will post more later as I try it out.
4. Add gypsum to your compost if you are working on it at this stage. I am having a terrible time locating any right now!
5. Heat will kill the spoor. If the compost is generating much heat as it decomposes it is too soon to put the spoors in.