Looks like there's no solid answer on whether local honey helps.
I was curious about the methodology but didn't find a free version of the whole paper; I did, however, find a Master's thesis that evaluated both the study you linked to as well as a more recent one, and the overall results are inconclusive. In addition, the earlier study only tracked participants for 30 weeks. Meanwhile, allergy shots are generally expected to take longer to have a noticeable impact: the Mayo Clinic says that allergy symptoms "usually improve during the first year of treatment, but the most noticeable improvement often happens during the second year."
Nature Wanderer makes a good point about the different types of pollen, however. A super common allergen in the Bay Area is mountain cedar, which propagates over wind rather than relying on bees.