Consider a standard Brownian motion $B$. Define $B_\mu=(B(t)+\mu t)_{t\ge 0}$ for every $\mu\in\mathbb{R}$ and denote $\overline{B}_\mu(t)=\sup_{s\in[0,t]}B_\mu(s)$ for any $t\ge 0$.
Question 1. Is it possible to simulate $(B(t),\overline{B}_{\mu_1}(t),\overline{B}_{\mu_2}(t))$ (or even $(\overline{B}_{\mu_1}(t),\overline{B}_{\mu_2}(t))$) for $\mu_1\neq\mu_2$ with uniformly bounded expected runtime (over the choice of parameters $(\mu,t)$)?
Question 2. Let $\tau_\mu(t)=\inf\{s\in[0,t]:B_\mu(s)=\overline{B}_\mu(t)\}$ (for any $t\ge 0$ and $\mu\in\mathbb{R}$) be the time at wich the Brownian motion attains its maximum. Is it possible to simulate the tuple $(B(t),\overline{B}_{\mu_1}(t),\overline{B}_{\mu_2}(t),\tau_{\mu_1}(t),\tau_{\mu_2}(t))$ for $\mu_1\neq\mu_2$ with uniformly bounded expected runtime?
Note that it is possible to express (in a not very tractable way) the tail proability $\mathbb{P}(\overline{B}_{\mu_1}(t)>x_1,\overline{B}_{\mu_2}(t)>x_2)$. Indeed, suppose that $\mu_2>\mu_1$ and let $T_\mu(x)=\inf\{t>0:B_\mu(t)=x\}$ be the hitting time of $x$. By hypothesis, conditional on $T_{\mu_1}(x_1)=s\in(0,t)$, the events $\{\overline{B}_{\mu_2}(t)>x_2\}$ and $\{\sup_{u\in[s,t]}B_{\mu_2}(u)>x_2\}$ agree a.s. for any $x_2>x_1>0$. Thus, the strong Markov property gives $$\begin{split} \mathbb{P}(\overline{B}_{\mu_1}(t)>x_1,\overline{B}_{\mu_2}(t)>x_2) &=\mathbb{P}(T_{\mu_1}(x_1)\le t,\overline{B}_{\mu_2}(t)>x_2)\\ &=\int_0^t \mathbb{P}(\overline{B}_{\mu_2}(t)>x_2|T_{\mu_1}(x_1)=s) \mathbb{P}(T_{\mu_1}(x_1)\in ds)\\ &=\int_0^t \mathbb{P}(\overline{B}_{\mu_2}(t-s)>x_2-x_1-s(\mu_2-\mu_1)) \mathbb{P}(T_{\mu_1}(x_1)\in ds). \end{split}$$
The quantities in the display have explicit formulas (see e.g. Devroye's paper, which unfortunately does not have the answer to my questions) and can thus be 'computed'. A similar argument works for the vector $(B(t),\overline{B}_{\mu_1}(t),\overline{B}_{\mu_2}(t))$, however, in neither case can I actually get to a simulation algorithm. The distribution seems to become too complicated for that.
Helpful references are also appreciated!