Thanks to the discussion with Glen, I realized that (assume ${X_a}$ is the number of a's and ${X_b}$ is the count of b's):
$${X_L}|{X_M} \sim B\left( {n - {X_M},{p_L} = \frac{{{\theta ^2}}}{{{\theta ^2} + {{\left( {1 - \theta } \right)}^2}}}} \right)$$$${X_a}|{X_b} \sim B\left( {n - {X_b},{p_L} = \frac{{{\theta ^2}}}{{{\theta ^2} + {{\left( {1 - \theta } \right)}^2}}}} \right)$$
Hence:
$$E\left( {{X_L}|{X_M}} \right) = \left( {n - {X_M}} \right)\frac{{{\theta ^2}}}{{{\theta ^2} + {{\left( {1 - \theta } \right)}^2}}}{\rm{ }}$$$$E\left( {{X_a}|{X_b}} \right) = \left( {n - {X_b}} \right)\frac{{{\theta ^2}}}{{{\theta ^2} + {{\left( {1 - \theta } \right)}^2}}}{\rm{ }}$$
Making:
$${\mathop{\rm cov}} \left( {{X_L},{X_M}} \right) = - 2n\left( {1 - \theta } \right){\theta ^3}$$$${\mathop{\rm cov}} \left( {{X_a},{X_b}} \right) = - 2n\left( {1 - \theta } \right){\theta ^3}$$
Leading to:
$$V\left( {{{\hat \theta }_{MLE}}} \right) = \frac{{\theta \left( {1 - \theta } \right)}}{{2n}}$$
:)