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I worked on a vessel that was (and is) state registered (Maine, complete with numbers/decal port and starboard), and was also CG Inspected (with a COI). I was aboard the day we had the inspection, and I don't believe it was a "courtesy" inspection - I thought it was required in order to keep our rating (?). If I remember correctly it stipulated number of passengers, where we could operate, etc. (I may be wrong, and those stipulations may originate somewhere else, but I was aboard for the CG Inspection).
I don't know how common this is, but I used it as the basis for thinking all four combinations must be possible.
If it has a COI, it will be in the PSIX. Look up the Maine vessel in the PSIX and see if it is documented. Spree is documented as well as state registered. I wonder if ferries owned by municipalities (Casco Bay Ferries) are state registered or documented?
Edit - they are documented.