Hopefully, you know that LCBR is in South Hole Sound, which is pretty much sand & eel grass...not what most divers would consider for a shore dive. However, some do take a rowboat/kayak out to Owen Island and snorkel out there.
Most people are probably going to want to have a rental car to haul their gear, which traditionally have been very expensive on Little Cayman. Its been a few years since I've been to LC and rented a car, but I want to say it was in the ballpark of $100/day (after including gas)...it makes for a good argument to pay for the afternoon diveboat instead
For shore diving spots, there's a rugged beach entry on the (usually windward) southside, just beyond Pirate's Point. Haven't done it myself, but I've walked that area of beach and the waterline is a pretty far distance from where you can park (car or bike).
There's also possibly the Salt Rocks pier (close parking; watch for "No Diving" signs and if there's a ladder for exit .. the concrete stairs look too dicey for my tastes). There might still be a ladder at Splash House (basically next store; private property). The road from LCBR to here is paved and isn't a bad bike ride...nice place for a sunset, mosquitos willing.
A friend used to go to Sam McCoy's frequently and he would shore dive out of there someplace right out to BBay Wall...I'm not sure exactly how/where, or if he was going up the road to Cumber's. Further east on the northside, there's a good beach entry/exit at Cumber's Caves that's IIRC down a dirt road/path from the main road...I know that I've heard of some people from LCBR using the resort bicycles to come over here for a snorkel, and I think that the cross-island road got paved a few years ago, so this might not be a bad bike ride to try today.
Finally, one could drive all the way out to Point of Sand, but I'd probably consider that to be more of a snorkeling site too and more of a 'special trip' to take for reasons more than just to get one's face wet. There's probably also some sites along the north shore to the east of Jackson's point, but there's a fringing reef that picks up in this area that probably limits much of it to the shallows inside the fringing reef.
There's some Little Cayman homeowners on SB; hopefully they'll notice this thread and provide some signficantly better advice.
-hh