Based on your location and the potential for water infiltration into the walls, I would not recommend installing batt or cellulose within the wall cavities.
Also with that said the approach of installing spray foam insulation through holes will not guarantee proper and even coverage within the wall cavities.
- Spray foam insulation emits water vapor during the curing process. By encapsulating the foam within the wall cavity this vapor could have a tendency to collect and given a food source for mold growth will allow mold to grow with in the walls.
- Be careful with soy-based foams. In recent reports and discussions with local contractors, the soy-based foams do not properly expand and are more susceptible to weather conditions during the installation process.
Because you are in a zone 5 for hurricanes, you have the chance of seeing 157 +/- wind speeds and driving rains. Being that your home was built in 1920 and is an uninsulated beach house, these winds speeds can force rain into the internal cavity of the walls. Having a spray foam insulation installed would eliminate the driven rain from entering the home, however could allow it to become trapped between the exterior siding and insulation allowing rot and deterioration to happen in the exterior siding/walls of your home.
Since you are planning on remodeling I would highly recommend that great attention is given to the water/weather tightness of your home specifically at locations around windows and doors. On a recent project in Florida, we had performed and observed high pressure water tests at each window and door for potential leaks at the interior with thermal imaging cameras. This test is easy and can be performed by your contractor prior to closing up the interior.
Since the interior is tongue and grove bead board, with care it can be removed, labeled and reinstalled to allow the insulation to be properly installed and off gas vapor during the curing process prior to encapsulating it in the walls. For the insulation I would recommend using a closed cell foam in a minimum thickness of 2 inches. This thickness will act as a vapor barrier and provide an insulation value of an R-16 +/- in the walls.
For the upstairs, insulation between the rafters creates a hot deck and will turn the attic space into conditioned space allowing the heat to rise above the interior spaces increasing the comfort of the occupants while increasing the efficiency of the HVAC system.
Taking the proper preventative measure now will increase the durability and livability of your beach home. Seek services and advice from professionals familiar with your demographics and weather patterns to recommend the proper solutions. Look at the 'Find a Pro' section of this website to locate Architects, Builders and Energy professionals in your area.
Good luck and many years of enjoyment of your beach house.