Solution:

Summing the geometric series, the numerator is $4^a\dfrac{4^{k(m+1)} - 1}{4^k - 1}$ and the denominator is $2^a\dfrac{2^{k(m+1)} - 1}{2^k - 1}.$ Using $4^N - 1 = (2^N - 1)(2^N + 1),$ the ratio simplifies to $$2^a \cdot \frac{2^{k(m+1)} + 1}{2^k + 1} = 964.$$

Since $964 = 4 \cdot 241,$ take $a = 2,$ so $\dfrac{2^{k(m+1)} + 1}{2^k + 1} = 241.$ With $k = 4,$ $241(2^4 + 1) = 241 \cdot 17 = 4097 = 2^{12} + 1,$ so $k(m+1) = 12$ and $m = 2.$

Then $a + k + m = 2 + 4 + 2 = 8.$

Thus, the correct answer is A.