The real problem with this is not the doctors freedom of speech. The govt controls MANY of the things the doctor says. There are a myriad of ways doctors change their behavior and either say things or don’t say things because of govt regulation.
The doctors who ask this are asking for one reason, because they are anti-gun. They are ok protecting their rights but are not ok with you protecting yours.
Being creative about not lying or deciding to lie are work arounds.
Reading your post, I just thought of another great response if you have an anti-gun zealot doctor asking the questions:
I’m sorry Dr., but talking about guns when I’m here for a sprained ankle kind of gives me the creeps.
I really don’t care what premise they use, I like to use my “play dumb” card in these situations rather than my “forcefully exercising my rights” card. It goes a lot farther.