I believe HA will charge you their standard inter-island baggage fees since you're not flying HA "international" flight in conjunction with your inter-island flight. It will make no difference with HA that you're *G member or flying LH/UA in F. Your inter-island flight is separate and HA will charge you bag fees.

Sorry to disappoint you but I'ld be very surprised (actually shocked) if HA waived your inter-island baggage fees just because you're *G or fly UA/LH "F".

In your situation, no matter who you fly inter-island you'll be subject to baggage fees (not unless you purchase HA "F" inter-island fare which includes 2 free checked bags according to HA website baggage fees/policy web page).

KOA-OGG both HA and WP (Island Air) has nonstop flight
OGG-LIH WP has multiple nonstop flights and HA most if not all connect in HNL

Of course there's always YV but in my opinion I'ld stick with either HA or WP