Is it permissible to combine the Shafi'i and Hanbali madhhabs regarding the issue of ghusl, such that one performs ghusl with the intention of wudu', rinses the mouth and nose, and then immerses oneself in still water, or is this not permissible?

The answer is summarized as follows: The matter is easier than you imagine. You can follow the Shafi'i madhhab (definition) and rinse your mouth and nose with the intention of performing ablution, then immerse yourself in the water. As for the Hanbali madhhab, it does not permit this, nor does it permit immersion followed by rinsing the mouth and nose. This is because the condition of sequence would be missed, and also, according to them, used water is pure but not purifying. To reconcile between the two madhhabs, you can rinse your mouth and nose first, then immerse yourself in the water. By doing so, your ritual impurity will be removed according to those who permit this scenario.