I don't think I was clear in my post. F3 is not "on" first base. F3 would be 10 feet off first base in the base path between first and second. As soon as the batter/runner continues, F1 throws to F3, F3 tags the batter/runner and then throws to F2 if the runner on third is going home. It's pretty much the same idea, but with a shorter throw.
Now you are running the risk of F3 being called for obstruction. If F3 does not have the ball and impedes the runner in any way, they would be guilty of obstruction. Based on your description, I very likely would have an obstruction call on F3.