Thanks to all who commented on this, and lots of thanks to Wishgranter who created a series of orthos which could be stitched to a full unwrapped image.
I went back to this issue about two weeks ago and decided I had to solve the whole problem again, mainly because it was extremely difficult to stitch the orthos in a way that would not distort the scale of the image or cause other problems. I now want to share with you how this can be done.
I decided to create an image not by unwrapping the texture but by unwrapping a dense point cloud, so I first exported a dense point cloud at ultra quality (883 million points, 36 GB including RGB colour). I also created a lower-resolution model (.ply) and used MeshLab to pick the positions of a series of pairs of approximately vertically aligned points. Then I started writing code: the point pairs are used to compute an average vertical vector. This is then used to rotate the dense point cloud accordingly. Then, the mid-points of the point pairs are projected to the now horizontal xy plane, creating a polyline along the sediment profile. Now, the distance of a point of the rotated dense point cloud from the new xy plane is the image Y coordinate, the distance along the polyline is the image X coordinate, the distance from the polyline (projected into the rotated xy plane) is a "DEM" type value, and the image xy grid is filled with the point RGB values. Two GPS points (averaging each thousands of one-second measurements) together with their positions in the model are used to scale the image grid. After two weekends and a number of evenings of thinking and programming and testing, the programme needed about 6 hours to go through the 883 million points, and then I finally had a metrically correct unwrapped image!