I intend to demonstrate correlation under the parameters you mentioned - easier access to nutrient poor foods, less access to indigenous food sources, consumer behavior, and increased severe weather events (periods of drought, flooding, etc.). Like you said it's hard to use global temp, however ocean temps have been increasing and are expected to continue to increase - leading to further acidification of the reefs surrounding the pacific islands, which affects mostly the on-shore fishing industry(which stays on the islands basically). The off-shore fishing industry (which is exported) is affected mostly by cyclical weather patterns such as el niño, la niña, so I'm not going into that. The majority of my project actually looks at cultural practices and adaptation, and is mostly qualitative. Lots of surveys and interviews with industry representatives and highland and lowland communities on a subset of 5 island groups that I chose based on input from the Fiji Ministry of Health. It shouldn't be terribly difficult to demonstrate there are linked problems - the goal of the project is mostly to tap into the way that communities that have been forced to adapt are doing so (or aren't), and possibly project some rates at which the health care system will be further over-burdened in the future and identify and communicate more widely culturally appropriate and feasible ways to help communities adapt to the changes in their environment in a healthier way.
I should note I'm not a climatologist - I'm an epidemiologist, but I'm working with some excellent climate people at USP.
There are a number of reports that touch on this and its many factors - if you're interested message me your email and I'd be happy to link them to you. I've been working in Australia on this and with a few groups in the Pacific, so they send me stuff on a regular basis.
So glad to hear you find it interesting!! I could talk about it forever, haha.