A Frenzel with regulator in-mouth is called a Marcante-Odaglia. It is slightly LESS effective than original Frenzel (that is done by free divers, with the mouth closed).I only tried the frenzel dry so far and tomorrow I gonna try it in the water. Hope it works with regulator on mouth.
You should practice performing both techniques, as removing the reg from the mouth for performing pure Frenzel is definitely unpractical when scuba diving...
Albeit in some case it is necessary, exactly because Frenzel provides more pressure than Marcante-Odaglia, so when a tube has difficulties opening, you remove the reg and perform pure Frenzel, for maximum pressure.
This is what is done usually by free divers, when they use a Scuba system: they are used to pure Frenzel, so they feel the need of removing the reg from the mouth for performing it properly.