MapToSurface / Sphorph with trimmed surfaces

I’m having some difficulties in finding the right approach in spatial deforming a surface and projecting curves along with it. In the image you can see the original surface (red) and the new one (green highlighted). Ideally it would work like cage-editing (with point deformation), but I can’t get that to work. Spatial deformation works almost as I want, but I couldn’t get the fall-off linear.

Converting the boundary to a polyline, modifying the vertexes and sporphing the curve gives a better outline definition, but the projection is not working as I hoped. (The surface control points are not modified like I hoped).

Some context:
The original surface consists of a user defined outline, which is split in multiple surfaces. Each of the individual surfaces need to be adjusted on several points along the edges and the corners. (27.7 KB)