Even ungrounded systems are required to have a grounding system and bond noncurrent-carrying metal parts. They are also required to have a ground detection. With other grounded systems on the premises, the grounding systems should be coupled IMO but as SDS's, the NEC does not require such unless part of the same building or structure. So with that said, I still agree with your inspector... and add that there needs to be ground detection installed somewhere if the secondary is truly ungrounded. Whoever engineered this project should know that. I do not know the circumstances why they decided to go ungrounded, but as I see it, the secondary should be grounded, if not solidly, at least go with high impedance grounding, especially if there will be no one to monitor ground detection on a frequent basis.
PS: the other system, especially the grounding transformer part, sounds weird. Can you post the one line (please verify non-proprietary status and/or censor the qualifying data before posting).