This is incorrect advice. US law, effective for the most part in January and in full in July, abandons the entire IATA MCT complexity. The text of the new rule is below and can be found at 14 CFR Sec. 399.87. The issue for OP is to determine whether he purchased separate itineraries for HNL-OGG-HNL. If so, he must pay the fees associated with the inter-island itineraries. If the inter-island segments are part of the LAX-HNL-LAX itinerary, whatever his baggage allowance is for LAX-HNL (1st segment of the 2nd itinerary) will apply to the remaidner of that itinerary.

In practical English, if the itner-island is not part of a UA codeshare, he is likely on yet a third itinerary, even though he probably booked everything at the same time (hence the term "booking" is no longer used).

The simplest way to sort this out is for OP to call Island Air and ask why he is not receiving the benefit of the UA itinerary. Their answer is not the end of it, but they might simply agree and note the PNR.

§ 399.87 Baggage allowances and fees.
For passengers whose ultimate ticketed origin or destination is a U.S. point, U.S. and foreign carriers must apply the baggage allowances and fees that apply at the beginning of a passenger's itinerary throughout his or her entire itinerary. In the case of code-share flights that form part of an itinerary whose ultimate ticketed origin or destination is a U.S. point, U.S. and foreign carriers must apply the baggage allowances and fees of the marketing carrier throughout the itinerary to the extent that they differ from those of any operating carrier.