Oh i'm so sorry to hear about your twins.
It takes such courage to try again when you have had such terrible losses, you are very brave.
I think your history means the GP must refer you, even if the current consultant says no to more tests.
Perhaps look around for the best place, rather than the nearest?
(the PM on my son revealed clotting issues, so i'm now at a hosptial ages away where they have specialists for this sort of thing, the local consultant didn't know a lot and even though i'm out of their catchment area, I explained to the GP why i thougth it was the best place for me and they agreed to refer me)
I don't know much about the immune factors, perhaps one of the MN on the thread linked above could help, but would your DD and the neonatal death indicate that it is unlikely to be immune (my impression, perhaps totally ignorant and misinformed is that if you have this condition you tend to lose all babies early in the PG??)
I'm wondering if the recurrent miscarriage unit at st mary's in london might be the place to push for- its a european centre for excellence and although i have no experience of it myself, MNers who have manage to get referrred there speak very highly of it. Worth a google, or starting a thread to ask for experiences, maybe?