Consider a Brownian motion $B=(B_t)_{t \ge 0}$ as well as a positive random variable $\xi$ independent of $B$ and a measurable, locally bounded function $u: \R \times [0,\infty) \to [0,\infty)$. Let $$\tau:= \inf\left\{T \ge 0: \int_0^T u(B_s,s) \D s \ge \xi\right\}$$ be the first time the time-inhomogeneous additive Brownian functional associated with $u$ reaches the threshold $\xi$. We will analyze the asymptotic behavior of $\p(\tau \gne T)$ as $T \to \infty$ and, in particular, provide sufficient criteria for this probability to decay like a multiple of $\frac{1}{\sqrt{T}}$. Subsequently, we will discuss the existence and long-term behavior of the associated conditioned process, i.e., of $B$ conditioned on the rare event $$\{\tau=\infty\} = \left\{\int_0^t u(B_s,s) \D s \lne \xi \text{ for all } t \ge 0\right\}.$$ Our framework, in particular, covers occupation times below any moving barrier dominated in modulus by $t^\gamma$ for some $\gamma \lne \frac{1}{2}$ as $t \to \infty$. Further, it covers the case where $u$ is a modified solution of the FKPP equation. This will be the key to upcoming results concerning branching Brownian motions with critically large maximum, a joint project with Bastien Mallein (Toulouse).