Let $\varphi$ be a smooth function such that $\varphi(x)=0$ for $x\leq 0$ and $\varphi(x)=1$ for $x\geq 1$, and $0\leq\varphi(x)\leq 1$ else. If $\tau_1$ and $\tau_2$ are is the first times $B_t$ hits $1,2$ respectively, and $M$ is a maximum of the $B_t$ on $[0,1]$, put $F=-M\varphi(\frac{x-\tau_1}{\tau_2-\tau_1})$. Then, $\sup B_t+F\leq 2$ almost surely, hence $\mathbb{P}$ is not absolutely continuous with respect to $\mathbb{Q}$.

Let $\varphi$ be a smooth function such that $\varphi(x)=0$ for $x\leq 0$ and $\varphi(x)=1$ for $x\geq 1$, and $0\leq\varphi(x)\leq 1$ else. If $\tau_1$ and $\tau_2$ are is the first times $B_t$ hits $1,2$ respectively, and $M$ is a maximum of the $B_t$ on $[0,1]$, put $F_t=-M\varphi(\frac{t-\tau_1}{\tau_2-\tau_1})$. Then, $\sup (B_t+F_t)\leq 2$ almost surely, hence $\mathbb{P}$ is not absolutely continuous with respect to $\mathbb{Q}$.