Are your periods still regular? As you say, they are heavy and making you anaemic. And anaemia can cause a lot of the problems you are experiencing.
Has your GP offered you treatment for heavy periods? Such as a Mirena coil?
Taking iron is treating the symptoms but not the cause.
It doesn't matter that you have never had panic attacks or waking up before. What does that prove? Only that you weren't in peri menopause.
You sound as if you have some anxiety but please believe that there are many women like you who felt terrible, were convinced they had all sorts wrong, and all along it was menopause.
The only way to find out is to try a low, starting dose of HRT.
If your symptoms improve, then that's the answer.
Please take no notice of the nurse.
Anyone who thinks it can't be menopause, because you aren't having flushes, really knows nothing. I wonder how many other women she's misinforming?
Listen to @Stroopwaffels and think about joining some online support groups and you will find lots of other women like you. I think you are underestimating how peri can affect some women and imagining it's something worse.