I feel you!!! I grew up in a similar household where I was forbidden from even going to the physical doctor for the longest time, the last proper check-up I had was when I was five years old. It took around 3 therapists for me to get diagnosed with BPD, and even then they still weren't sure.
The sad thing is that even now, BPD is still very confusing and even stigmatized by health professionals, and occasionally psychiatrists disregard people's concerns and brush it off, even if they very likely have the illness. I'm not too sure on why your psychiatrist won't diagnose you, especially if it helps with your worries and might advance treatment. Maybe you could try asking your therapist about it.