How to deal with $\mathbb{E}\left[\left(\int\limits_0^tW_u^2 u^2\partial u\right)^2\right]$.
I have limited time just now, and can fill in more details later if you want. Idea comes from here $$\Bbb E[( \int_0^t W_u^2 u^2 d u)^2 ] = \Bbb E[ \int_0^t W_u^2 u^2 d u \cdot \int_0^t W_s^2 s^2 d s ] = \Bbb E [\int_0^t\int_0^t W_s^2 W_u^2 s^2u^2 d u ds ]\\ =\int_0^t\int_0^t \Bbb E[ W_s^2 W_u^2 ] s^2u^2 d u ds $$ and
$$E[ W_s^2 W_u^2 ] = E[ W_{\min (s,u)}^2 (W_{\max(s,u)}- W_{\min (s,u)} + W_{\min (s,u)})^2 ] \\ = \Bbb E[W_{\min (s,u)}^4] + \Bbb E[W_{\min (s,u)}^2]\Bbb E[(W_{\max(s,u)} -W_{\min (s,u)})^2], $$ which can be further and explicitly computed.