Let $A\in\mathbb{R}^{m\times n}$ and let $B\in\mathbb{R}^{n\times n}$ be such that $\alpha I\preceq B\preceq\beta I$ for some $\beta\geq\alpha>0$ (here, $I$ refers to the identity). I wonder whether there always exists a constant $\gamma>0$ such that the matrix inequality \begin{equation} A^{\top}AB+BA^{\top}A\succeq\gamma A^{\top}A \end{equation} holds? Clearly, the assertion is true whenever $\alpha=\beta$.
We may also assume that $A$ has full row rank (rather than full column rank).
No. Let $$ P=\pmatrix{5&2\\ 2&1},\ A=P^{1/2},\ B=\pmatrix{1\\ &3}. $$ Then $A^\top AB+BA^\top A=PB+BP=\pmatrix{10&8\\ 8&6}$ is indefinite and hence it cannot be greater than or equal to the positive semidefinite matrix $\gamma A^\top A$ in positive semidefinite partial ordering.