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):

$${X_a}|{X_b} \sim B\left( {n - {X_b},{p_L} = \frac{{{\theta ^2}}}{{{\theta ^2} + {{\left( {1 - \theta } \right)}^2}}}} \right)$$

Hence:

$$E\left( {{X_a}|{X_b}} \right) = \left( {n - {X_b}} \right)\frac{{{\theta ^2}}}{{{\theta ^2} + {{\left( {1 - \theta } \right)}^2}}}{\rm{ }}$$

Making:

$${\mathop{\rm cov}} \left( {{X_a},{X_b}} \right) =  - 2n\left( {1 - \theta } \right){\theta ^3}$$

Leading to:

$$V\left( {{{\hat \theta }_{MLE}}} \right) = \frac{{\theta \left( {1 - \theta } \right)}}{{2n}}$$

:)