In my experience, the fireworks are not during the first 2 weeks of June. I think they start with the summer season after june 10th or something like that. THe fireworks are not part of Disney, but of the Shelter Cove Merchants Association. (Which Disney HHI Resort is a member) I hope you have better luck than me. I have been a member of HHI since 1998 and I have yet seen the firework show. I have been during firework season about 5 times and it has rained on everyone of those Tuesday evenings. They never did reschedule or They did and I was in SAV. Also, I will be there from the 5th-10th. So based on my track record of tuesday night rain, don't get your hopes up. Hopefully I can get some pixie dust and not bring a tues. night thunderstorm with me this time.
You can see the fireworks barge on the backside of the resort, sitting in the marsh.