They will probably not interfere with each other unless you're taking huge doses of Lugols. I would take Miracle-Mineral-Supplement at night and Lugols in the morning. Both Lugols or Miracle-Mineral-Supplement can be applied externally, directly to feet also.
Other wayse of killing athletes foot.
1. Heavily ozonated olive oil available on the net. Rub into feet like an ointment.
2. Ionic Silver. Take orally and soak feet in it. Get a machine to make it practical. silver only works when wet though. Possible to make a silver gel with products on this site.