There are ASTM and perhaps AASHTO (for your site) standards that apply to this form of testing. Before you go too far, I'd bring on board an experienced geotechnical engineer (even though you may be well experienced) and develops a program as if you were going into court It may be a good idea to bring on a lawyer, but be sure he (she) is construction experienced.
I suspect that a nuke gage was used on the job (the lazy way to get a lot of test data) and many swear by that. However, the standards tell how you calibrate that against known acceptable testing methods to be sure the work is done right.
There is another current thread getting extensive remarks (It may be in the "testing" room). They deal with this general subject and there are several experts there.
The only thing I would add about using the sand-cone method for calibration of the nuke, would be that the sand cone size has to be the large size (I believe it is 8" diameter), since that method also can have errors and you need a goodly large test sample to minimize those errors.
In my experience (I'm 81 years old and still working some), I was one of the first to use nuclear methods at Cornell University in grad school where it was developed in 1954,and I followed that through with equipment from NuclearChicago and now Troxler. You should have seen the crude first Troxler equipment. It is much improved now, but the general method is quite touchy.
Anyhow, as I then worked on many a job, it became apparent that if the nuke gage was off by anything, it was low on density. That is the reason why good calibration for your soil is necessary. In this case, you may find that to be one of the factors that affected you. Was the equipment calibrated for on-site soil, not some lab bucket of concrete.
Lab testing (Proctors), can be discussed at length with your problem, but you need standards that are generally accepted by the "industry", such as via ASTM to hang your arguments on.
Finally, of course, was the completed job performance deficient in any way? In other words did they really need 95 percent compaction? Of course some will argue these details, but a lot of work has been done that works out fine for end result and the 95 percent was an unrealistic thing to ask for. I have accepted compaction on some parking lots at 85 percent on clays and everything worked out fine. That 95 number also might be argued.
Good luck