You should be able to get your LC350 in and get a logbook, as there is no major difference between it and the 1980 cut-off year LC350 (1982 in England and NZ, time to re-examine ?), however an RZ350 is not acceptable. It'd be worth trying to get hold of an unarguably-legal 1980 chassis to help things through, possibly. Go to the Island Classic this weekend and ask around...