Let $x$ be the total number of points and $y$ be the desired answer.

Then from the problem statement we get that $$\dfrac{x}{4} + \dfrac{2x}{7} + 15 + y = x.$$

Simplifying yields $$y + 15 = \dfrac{13x}{28}.$$

We know that $y \leq 14$ since none of the $7$ team members scored more than $2$ points.

We also know that $x$ must be a multiple of $28.$ If $x = 28,$ then we get that $$y + 15 = 13 \Rightarrow y = -2,$$ which is not allowed.

If $x = 2 \cdot 28 = 56,$ then we have that $$y + 15 = 26 \Rightarrow y = 11,$$ which works.

The next possible total is $x=84,$ which would give $y=24>14.$ Every larger multiple of $28$ makes $y$ still larger, so $y=11$ is the only possible value.

Thus, the correct answer is B.