If you don't need a fast tele sports lens for what you do and the inherit shallow DOF of longer focal lengths is enough with f4 for your purpose, then go with a lighter and less expensive f4 lens.
The lighter weight option might make it an easier choice to take it with you for travel or the occasions where you thought you would not need it but take it anyway.
If you think you will miss or compromise shots without f2.8 then it is a matter of deciding to what degree this will happen vs the cost, weight and bulk of the f2.8.
The f2.8 does have versatility in its favour if you shoot a wide range of subjects and Plays with Light suggestion of combining one with a teleconverter will negate any weight penalty if you would otherwise want to have a longer lens also.
Who would have thought one stop could raise so many complications.
FX vs DX body?
One stop of extra ISO is less likely to be an issue on an FX body if you chose a 70-200 f4.
A DX will of course give more reach, I which case you might not need a longer lens or teleconverter.
As with all these gear questions it comes down to what you shoot.
There is never an easy answer and what ever you get, you will learn to adapt to minuses and enjoy the pluses.