I think that you would get some pieces of ice containing protein this way, so I doubt it will be tasty or useful in terms of cooling down on a hot day. The problem is that the ice-cream is still so creamy after being frozen because it has so much fat in it, and if you won't have this fat in your protein ice-cream, it will be just a piece of ice.
If you will add this fat into the ice-cream, then the use of protein will be kind of pointless. The overall idea is good, but unfortunately I don't see how it can be executed.