Tim isn't around this week, but I can identify the problem. With Livemap, you can get a route between 2 points which you select on the map. The script doesn't allow you to select discrete points. It takes the segments you have selected and calculates their mid-points: these two points are then passed to Livemap to return a route.
I think that there is some very slight difference in the way the GPS coordinates are calculated in WME and Livemap. This occasionally results in issues like this - where a coordinate passed to Livemap is actually closer to another segment. I find it frequently happens with split carriageway segments which are long and curved.
The general workaround for this is to try and select earlier or later start/end segments on the same route. In this case, my gut tells me that trying this segment
as the destination might work better - assuming I've followed the route you're trying to test.