How to store fish bones?
If you produce edible food, it cannot be left at room temperature indefinitely. The bones in ossuaries are not cooked, softened fish bones, they are only the mineral part of animal bones, with the organic part decayed away. So there is no comparison there. Treat them like any other cooked food. 

There is nothing which gets firm after going through the freezer, stuff gets mushier if anything. For bones, there shouldn't be any change in texture. 

So the standard storage lifetime applies to them, just as for any other cooked meal: 2 hours outside, 3-5 days in the fridge, as long as you want in a -18 C freezer.