How to find the market equilibrium by surplus maximisation when there is a consumer subside?
never work on weekends :-)

What you really do here is to find the total surplus in terms of $Q*$ and then maximise it setting the derivative to $Q*$ equal to zero.
So, you just maximise 
$$TotalSurplus = - \frac{3}{16}Q^{*2}+\frac{9}{2}Q^{*}  + 2 * Q^{*}$$
for $Q*$ and you obtain the expected results.