The MV Keith Tibbets lies in the waters of Cayman Brac, some 90 miles from Grand Cayman. AFAIK, only the Cayman Aggressor liveaboard makes the trip by boat and only if the seas are calm enough. Some weeks in the winter months it never manages to make the crossing.
Air service to Cayman Brac is available on both Cayman Airways and Island Air. For a round trip in a single day, you'll have better luck on Island Air. Flying after diving is a bit of an issue with that plan, obviously. I do not know the normal altitude for the flight.
"(M)ore obscure, less touristy dives" is a bit of an issue. The least travelled path is in East End. The entire East End is well served by three dive operators, none of them are what I would call small. The west side is served by many more (30ish?) operators, some of which are small operators. The west side operators serve an area that includes the North Wall, the west side, and South Sound.
There is one stretch of shoreline that none of the dive operators regularly service. The south shore of Grand Cayman between about Frank Sound on the east to South Sound on the west is a bit too far for most operators to travel regularly. If you can find someone with a small boat on a trailer, they could probably launch at the Frank Sound boat ramp and cover quite a bit of that territory.