Ito Isometry for Brownian Bridge

Question

I have been looking at the following expression, and I'm a bit stuck: $$\left(\int_{0}^{t}\frac{dW_{u}}{1-u}(B_{1} - B_{u})\right)^{2}$$ where $\{W_{t}\}_{t\in[0,1]}$ is a standard Weiner Process and $\{B_{t}\}_{t\in[0,1]}$ is a generalized Brownian bridge (endpoints need not be the same). If I apply expectation to this, will the Ito Isometry hold? Is the resulting expression just: $$\mathbb{E}\left[\int_{0}^{t}\frac{du}{(1-u)^{2}}(u - u^{2})^{2}\right] = \mathbb{E}\left[\int_{0}^{t}u^{2}du\right]?$$ Any help or clarity is greatly appreciated.