Soapy water or Azamax drenches?
 

Any objections to using soapy water drenches to deal with a root mite infestation? (1tbsp/quart Dr Bronner's Peppermint Soap)

Alternatively, would soaking with a solution of Azamax suffocate the roots? I've heard Neem Oil, or other oil, drenches aren't recommended as they can obstruct the roots' gas exchange.

Oil shouldn't be used on roots for the reason you mentioned, and it shouldn't be used on above ground plant parts too often, for the same reason.

I don't know how peppermint oil affects plants. I would use plain dish soap, or an insecticidal soap made for that use.

Azamax is not oil-based.

Azadirachtin is the active ingredient in neem oil but is not an oil, itself.