 Solution

There is no density on the full infinite path space; the two laws are mutually singular. Under Wiener measure, $B_t/t\to 0$ almost surely. Under the shifted law, $(B_t+t)/t\to 1$ almost surely, so the same measurable tail event separates the laws.

For completeness the Brownian limit follows from the Gaussian bound at integers and the maximal bound on each interval $[n,n+1]$: for every fixed $\eta >0$, both ${\sum }_n\mathbb{P}(|B_n|>\eta n)$ and ${\sum }_n\mathbb{P}({\sup }_{0\le s\le 1}|B_{n+s}-B_n|>\eta n)$ are finite. The Borel-Cantelli lemma and continuity fill the gaps between integers. Intersect over positive rational $\eta$. This proves the limit and the claimed singularity, which is infinite-horizon singularity of Brownian motion with constant drift.

On every finite horizon there nevertheless is the density $e^{B_T-T/2}$. Its failure to give a global density is exactly the distinction required by the printed half-line path space.