hey...
I have been using the "roadkill" to Uv map this model.This soft is still in it's development stage.But it works fine.The only problem with this soft is u have to uv map the whole model at a time and u have to use edge to separate the model.

link https://www.pullin-shapes.co.uk/page8.htm

i have talked to the technical team of RoadKill.Thay said this problem will be resolved in the next version which will not be free.

....well, the problem ur facing is very weird.But i have made a curved surface too,then rigid bind it.Then placed it like the following picture.But I m not facing problwm like u!...But i have used cylindrical project instead of planed.And i think cylindrical is better rather then planer in this case, which will save a lot labor and stretching problem.