Depends on your age and your resources for doing another retrieval to obtain more embryos. If young and could cycle again, you could try with the prednisone at SG. For some people with immune issues, prednisone and high-ish dose fish oil are enough to get you through. If this is really your last embryo because you can't cycle again, you would be better off being under the care of someone who can properly manage things. Is your embryo PGS tested? If not, you could be looking at something that won't work genetically anyway but you won't know until you transfer and see what happens.
Could you just do a consult with someone willing to do the testing who takes a lot of insurances (Abbasi locally or SIRM in NY, for example)? Then you'll know definitively what your issues are. You could then take that information back to SG. Even if they don't "believe in" immune issues, you would feel more confident that what they can offer either is or is not potentially enough.