Is it possible to prove that this algorithm is big Omega $n^2logn$ time complexity?
Your recurrence can be written as:
$
T(n) = T(n/2) + f(n)
$, where $f(n) \in \Theta(n^2)$.
Then, by case 3 of the Master theorem, you have $T(n)=\Theta(n^2)$. This means that you won't be able to prove a lower bound of $\Omega(n^2 \log n)$.