I agree with various points by PP. We did immune dysfunction testing with a RI specialist. Most RE's do not subscribe to some of the theories. However, be careful what you wish for, or next thing you know, some over zealous RE / RI will be telling you you can never carry a pregnancy with your DH and you need a surrogate or sperm donor! I think somewhere in between disbelief and extremism would be great.
I think karyotyping is a good start for recurrent loss work up, also coagulation work up. Then you can get into the various autoimmune conditions, including thyroid. Then natural killer cells, HLA typing and so on.
I agree with PP that there may not be a really effective treatment for these issues yet. Not to mention, it may not actually be THE issue that is causing you problems. Still, I think it is probably worth doing.