Let $X$ be a Hausdorff separated locally convex space. Denote by $w$ the weak topology and with $\tau_M$ the Mackey topology on $X$ with respect to the duality $(X,X^*)$ where $X^*$ is the topological dual of $X$.
What can be said about $X$ if $\tau_M$ has a weak-compact neighborhood of the origin?
In other words there exist $V$ a $\tau_M-$neighborhood of $0\in X$ that is $w-$compact.
Maybe something in terms of (semi-)reflexivity.
You are on the wrong track: If a Hausdorff locally convex space has a bounded (and absolutely convex) $0$-neighbourhood $U$ it is already a normed space with the Minkowski functional of $U$ as a norm (this is one of the earliest locally convex theorems in the literature due to Kolmogorov). If thus the Mackey topology has a weakly compact (hence bounded) $0$-neighbourhood then the Mackey topology is the topology of a reflexive Banach space.