Actually you can. Just don't give them your SSN. I did this with Kaiser Permanente and it continues to work just fine.
If all you're doing is making them get the information from someone else, what exactly are you accomplishing? Adding a bit of friction to a process?
I mean, fine, if that's the aim, but it sounds like this was intended as more of a “they don't need this information at all, and if you don't give it to them, then they won't have it” technique, and (apparently) that doesn't work.
There is no "someone else" who will give the insurer this information.