Dividing the quadratic by $7 + 4\sqrt{3}$ to obtain a monic polynomial will give a linear coefficient of $\frac{2 + \sqrt{3}}{7 + 4\sqrt{3}}$. Rationalizing the denominator gives:
$\frac{(2 + \sqrt{3})(7 - 4\sqrt{3})}{7^2 - 4^2 \cdot 3}$
$=\frac{14 - 12 - \sqrt{3}}{49-48}$
$=2 - \sqrt{3}$
Dividing the constant term by $7 + 4\sqrt{3}$ (and using the same radical conjugate as above) gives:
$\frac{-2}{7 + 4\sqrt{3}}$
$=-2(7 - 4\sqrt{3})$
$=8\sqrt{3} - 14$
So, dividing the original quadratic by the coefficient of $x^2$ gives $x^2 + (2 - \sqrt{3})x + 8\sqrt{3} - 14 = 0$
From the quadratic formula, the positive difference of the roots is $\frac{\sqrt{b^2 - 4ac}}{a}$. Plugging in gives:
$\sqrt{(2 - \sqrt{3})^2 - 4(8\sqrt{3} - 14)(1)}$
$=\sqrt{7 - 4\sqrt{3} - 32\sqrt{3} + 56}$
$=\sqrt{63 - 36\sqrt{3}}$
$=3\sqrt{7 - 4\sqrt{3}}$
Note that if we take $\frac{1}{3}$ of one of the answer choices and square it, we should get $7 - 4\sqrt{3}$.
The only answers that are (sort of) divisible by $3$ are $6 \pm 3\sqrt{3}$, so those would make a good first guess. And given that there is a negative sign underneath the radical, $6 - 3\sqrt{3}$ is the most logical place to start.
Since $\frac{1}{3}$ of the answer is $2 - \sqrt{3}$, and $(2 - \sqrt{3})^2 = 7 - 4\sqrt{3}$, the answer is indeed $\boxed{\textbf{(D)}}$.