Discrete Time Fourier Transform (DTFT) cross correlation property
Your first formula in your question is generally wrong, that's why you can't prove it. The correct formula is

$$\sum_{n=-\infty}^{\infty}x[n]y^*[n]=\frac{1}{2\pi}\int_{-\pi}^{\pi}X(e^{j\omega})Y^*(e^{j\omega})d\omega\tag{1}$$

which is just Parseval's theorem.

If $x[n]$ and $y[n]$ are real-valued, $(1)$ can be written as

$$\sum_{n=-\infty}^{\infty}x[n]y[n]=\frac{1}{2\pi}\int_{-\pi}^{\pi}X(e^{j\omega})Y(e^{-j\omega})d\omega\tag{2}$$

which is the form you've correctly obtained.

In the special case that $y[n]$ is not only real-valued but also even, i.e., $y[n]=y[-n]$, then $Y(e^{j\omega})=Y(e^{-j\omega})$ holds, and $(2)$ turns into the first formula in your question.