I believe you will lower your chances of an upgrade if you have booked an accessible room. I don't know the exact numbers, but there are far fewer accessible rooms than non-accessible rooms, and thus fewer rooms that might be considered an upgrade. There are also a limited number of types of accessible rooms, as shown here:

Kauai Accommodations & Beach Hotel Rooms | Grand Hyatt Kauai

GH Kauai does block rooms a few days in advance of one's arrival. Trying to switch rooms 1 or 2 days prior to arrival will also lessen your chances of an upgraded room, as the better rooms are likely already assigned.