Chemical pregnancies are very hard, such a roller coaster. I had one in May and I'm really sorry you've had so many. I test the day after my missed period, because frequent testing makes me anxious, but some people feel the opposite and test earlier. You do you.
I definitely think you can push for your GP to investigate this. There are many investigations done for recurrent miscarriages (3 or more) and a chemical is still a pregnancy, so I'd push for this. They can check your thyroid, if you have any bloody clotting disorders, or certain antibodies.
Its also may be worth getting your progesterone checked, a hormone you need for implantation and to sustain the pregnancy. If your GP won't do it, but I'd push for it, then you can get a self administered test from Superdrug. From my tests, I was on the lower side. I'm no medic, but I thought that this could be one of the reasons for my chemical.
There are clinical studies that show that women with a history of loss should be given progesterone if they start bleeding. Some people, however, can take it straight after ovulation to encourage implantation and I think this is something the GP should be able to help you with. If not, you could go to a private gynaecologist, a few hundred pounds, rather than a few thousand for IVF! Worth a try, I'd say!
I was about to start IVF the beginning of last month, but surprisingly got pregnant naturally. I was worried about it being another chemical, but we got to 6 weeks but then I started lightly bleeding. The early pregnancy unit did a scan and I asked for progesterone, which the doctor gave me. I'm currently week 10 and I can't know for sure but I think the progesterone is helping. They normally tell you to take it up to 12 weeks.
Hope this helps! I really think GPs need to take women who are struggling with conceiving more seriously.