Oil is known to loosen up scale to the point where they will fall off, oil is really not recommended as far as mite treatment goes.
Drowning mites regardless of the method does not solve the problem in the long run, it will not take care of the eggs that have been laid and is only a quick fix that does not work in the long run.
2 advices from you
1# change vet
2# Get some PAM (provent a mite) or Black night to completely eradicate your mite problem, it will get rid of mite at all stages of development.
Note: I wonder if your vet would recommend soaking a dog in oil to get rid of fleas![]()