That is a significant amount of work to engage in for someone who is already in a compromised mental state :'(
They might (They do, this has happened to me, and the end result was not getting care I needed, but anyway) suggest you not inform them of your insurance, then it's your fraud, not theirs.
My wife is a therapist and tries to prevent this by being as transparent as possible. She posts her rates, her calendar, and makes it very clear that she is private pay only on her website. By the time someone reaches out to her, they usually know that it is going to work because they have had the opportunity to see all of that information.