How many samples are needed to reconstruct a path?
Here’s a simple idea: try a certain number $s$ of samples, and for each vertex, record how many times its colour agreed with the provided bit.

For vertices outside the path, they agree with probability $1/2$ on each iteration, while for vertices on the path, the probability is $1/2+1/2k$, where $k$ is the number of vertices on the path.

Thus, if $s\approx k^2\log n$, the Chernoff–Hoeffding inequality ensures that with probability at least $2/3$, all vertices outside the path agree for less than $s(1/2+1/4k)$ samples, and all vertices on the path agree for more than $s(1/2+1/4k)$ samples, which lets you pinpoint the path. If you don’t know $k$ in advance, you can just bound it by $n$.