Well, Frenzel can be performed only with the mouth closed.BTW, I know I said I used valsalva on the original post, but I used that term as it is more well-known. I actually use Frenzal from my freediving days, never Valsalva. Sorry to be misleading there.
Altough I have seen some free divers to use it while scuba diving (removing the reg from the mouth, equalising, and placing the reg in mouth again), I suppose that what you are currently using is some other technique, which is possible with the mouth open. Possibly Marcante-Odaglia (which uses the soft palate, but not the tongue as in Frenzel) or Toynbee.
Some divers with equalising problems develop their own personal method, often as a mixture of other ones.
However the name is not important, what matters is to open the Eustachian Tube and get equalisation.
If this is difficult in one of your tubes due to a long-term condition, you should evaluate a medical procedure for solving the problem permanently.
My wife also had problems, and her ENT suggested a balloon tube dilation procedure.
Luckily enough she solved her problem when she did follow the instructor course at the Nervi Training Center, where Duilio Marcante in person taught her how to equalise with the Marcante-Odaglia method.
But you are already beyonds this point, which was the second in my list.
So you are back to the first point: ask your ENT about possibilities for permanently solving your narrow tube problem.
You could also evaluate this DAN equaleasy workshop. One of my colleagues who had recurring equalisation problems got great improvement from it: DAN Europe - EqualEasy - course - DAN Europe - EqualEasy - course