I am trying to find a lower bound of the following form:
Given a vector $\left\Vert \boldsymbol{u}\right\Vert ^{2}\le1$ and given two non-expansion operators $\boldsymbol{A,P}$ (i.e., $\left\Vert \boldsymbol{A}\boldsymbol{u}\right\Vert _{2}\le1,\left\Vert \boldsymbol{P}\boldsymbol{u}\right\Vert _{2}\le1$) such that $\boldsymbol{P}$ is an (idempotent) projection matrix ($\boldsymbol{P}^{*}\boldsymbol{P}=\boldsymbol{P}$), if it holds that $$1-\epsilon \le\left\Vert \boldsymbol{A}\boldsymbol{u}\right\Vert ^{2} \\ 1-\epsilon \le\left\Vert \boldsymbol{P}\boldsymbol{u}\right\Vert ^{2}$$ for some $\epsilon\in\left(0,\frac{1}{2}\right)$, then we have that
$$\left(\boldsymbol{u}^{*}\boldsymbol{P}\boldsymbol{A}\boldsymbol{u}\right)^{2}\ge\,?$$
My intuition is that if $\epsilon$ is very close to $0$, then $\boldsymbol{A}\boldsymbol{u}, \boldsymbol{P}\boldsymbol{u}$ should be "close" with a high inner product.
I am also willing to restrict $\epsilon$ to a smaller interval if it helps.
Feeling like I'm missing something simple.
This isn't true, because you've only bounded the norm, which can't say anything about angles, which is what inner products are about. Let's take $u = (1,0)$ in $\mathbb{R}^2$. Then we can take $A$ to be a rotation matrix by any angle $\vartheta$ that you like, and $P$ can be the projection matrix onto the $x-$axis. Then $||Au|| = ||Pu|| = 1$, so your conditions hold for any $\epsilon$. On the other hand, these things can have any inner product you want, by choosing $\vartheta$ accordingly.