My initial thought is, since there's no real 'gradient' here, you probably need to brute force all possible rotations/positions. Perhaps you could quantise/downsample the map and path and brute force over that to find a shortlist of potential matches, then check each of these at higher resolution?

I would have thought it isn't possible, but apparently you can do significance tests of Random Forests (at least, this is suggested by rdrr.io/cran/rfUtilities/man/rf.significance.html). Maybe you can figure out what that method does and then transfer it to AUROC?