Your strict topology ($\beta_t$) is the strongest locally convex topology on $ C_b(T)$ coinciding with the compact-open topology on the unit ball $ \{ x \in C_b(T): \sup_{t \in T} | x(t) |\leq 1 \}$.

If each bounded linear functional on $ C_b(T)$
is $\beta_t$-continuous, then $T$ is compact. Suppose $T$ is not compact. Let $p\in\beta T\setminus T$ ($\beta T$ is the Stone-Čech compactification of $T$). Define the subset $N$ of $C(\beta T)$ by $N=\{f:0\le f\le 1, ~f(p)=1\}$. Labelling $N$ as $\{f_i : i\in I\}$ we make $I$ into a directed set by saying that $i\ge j$ iff $f_i\le f_j$. Let $g_i$ is the restriction of $f_i$ to $T$. Then the net $\{g_i : i\in I\}$ decreases pointwise to $0$. By the Dini theorem $g_i\rightarrow0$ uniformly on compact sets. By the above characterization of the strict topology we have that $g_i\rightarrow0$ in $\beta_t$. The linear functional $\delta_p(g)=f(p)$, where $f$ is the continuous extension of $g\in C_b(T)$ to $\beta T$, is bounded, so
$\delta_p(g_i)\rightarrow 0$. This is in contradiction with $\delta_p(g_i)=f_i(p)=1$.