Squaring both equations and adding gives $$\begin{aligned} &9 + 16 \\ &\quad {}+ 24\small(\sin A \cos B + \cos A \sin B) \\ &\quad = 37, \end{aligned}$$ so $24\sin(A + B) = 12$ and $\sin(A + B) = \tfrac12.$

Then $\sin C = \sin(A + B) = \tfrac12,$ so $\angle C = 30^\circ$ or $150^\circ.$ If $\angle C = 150^\circ,$ then $A \lt 30^\circ,$ making $3\sin A + 4\cos B \lt 6,$ a contradiction. Hence $\angle C = 30^\circ.$

Thus, the correct answer is A.