Nobody has seemed to mention that most DMEs, if not all, also show speed to beacon. Why not alter course 10deg and see if speed goes up or down.
If up, try another 10degs the same way and repeat.
If down, go 20 degs the other way and see what happens.
When you get two near enough the same, with distance decreasing, head half way between. Max speed = going straight to beacon.
Should work properly even with significant cross wind.
I thought the obvious answers of getting vectors from a radar unit or using GPS were avoiding the question - even if undoubtedly the best!
MikeJ