I had no problems with Kaiser when I applied 4 months ago. I just filled out the Kaiser form at the HPD window and waited the standard period before picking up my permit.
That's true of most Kaiser customers.
The problem arises when someone on the Kaiser clerical staff sends back information to HPD that flags the application. The patient might have had counseling 25 years ago for weight loss surgery, but because it was part of behavioral services, they flagged it. A physician ought to review these flagged applicants, but I don't know if that ever happens.
Once HPD puts a hold on the application, you have to provide a "clearance letter," stating you are "no longer" suffering from whatever behavior/mental thing they flagged you for and that you are okay for owning guns.
Kaiser is one of several HMOs who take the stance that they can't sign such a statement. So, going back to ask your PCP for a letter is useless. You have to pay out-of-pocket for another doctor to type and sign the clearance letter. They will likely require you to forward your medical records to their office.
Just another speed bump that I doubt ever accomplishes what the people running this shit show claim it does.