Hi, Simmers. Here is what I have experienced from Off The Grid gameplay with regards to cooking:
Use the Island Living Banana Leaves stoves. It will offer you far more options than the standard grills in the long run, particularly Island Dishes such as Pork Adobo and Island Vegetarian.
For a start, you can put a normal grill, just the gas can one or the silver one that are the cheapest; they work fine. My Sim gets Happy Moodlet eating food from those right from the start (my Sim is not a Foodie Sim).
All fridges work. I use Spa Day fridge.
You can put a stove that says "functions when off-the-grid". I use the one that has the wood underneath. But you get only two choices of food from the "cook with gas" option. So, I think you are much better off buying a banana leaves island grill.
In addition, the island grill gives you 7 or 8 servings per cook. You have to wait a bit longer but the food is ready when the banana leaves kind of disappears/look burnt. I have waited overnight before and it does not char the food at all! This means that you can go do your gardening, painting, photography, etc, while waiting for food.
Store the 7 servings in the fridge and your Sim can get leftovers every morning and night for the next few days.
If you are still into the modern day grill, you will get options such as Shish Kebabs, Hot Dogs, the usual.
If you have Outdoors pack, it is not worth it to get the bonfire unless you just want to see the graphics and increase the fun meter. Roasting fish and hotdogs takes too long. And Sims get an Uncomfortable Moodlet from poor food using the roast.
Plus, it is easier to go swimming to increase fun when you play in Sulani.
I would say, you can just get the cheapest fridge possible because it is used for storing only.