IO540,
I too wonder how it would work. Because in a real situation of you would go instantly to GPS (which you would have on anyway) and/or a radar unit (I am rarely not with a LARS), I took this as an academic puzzle. Turbulance is not allowed to the theorists.
Perhaps we should both try it next opportunity!
Nevertheless, the 'small angles' don't seem to be as limiting as you suggest.
At 10deg, the speed difference is 1.5%, which if cruising at 140kts, is over 2Kts down, and my DME is generally within 1Kt of my GPS ground speed, allowing for slant distance when close to the beacon.
Even better if the tests are done at 15deg course change. Just fly half way between those headings where you can observe a reduction. It should get you somewhere close.
Only theorising,
MikeJ