It is theoretically possible to do this but it involves some maths which need to be done with function nodes. I have not tried this yet, otherwise I would give you an example project.
I will describe the idea here in the hopes that you, or some of the node wizards here, can have a crack at it.
The general idea is to treat this as a transform of the texture space, where the texture space is in 3D coordinates. It's easier if the planet's centre is at 0,0,0. Transforming texture space in arbitrary ways can be done with a Warp Input Shader, a Vector Displacement Shader and some function nodes to do the necessary maths. For each point on the planet that you want to render (e.g. within your tile), the Vector Displacement Shader needs to produce a 3D vector which is the difference between the 3D point on the planet (within your tile) and the desired point in the whole spherical map (again in 3D) which the image map thinks it's using when you use the spherical projection mode. You'd use a Get Position in Texture node, do some maths, and subtract the original position from the output of your maths to get a displacement vector. For example, if you only wanted to shift the longitude of the map you could use a Rotate Y function node and then subtract the input from the output to get the difference (displacement), and plug that into the Vector Displacement Shader. To stretch the map over a different range of longitudes, or change latitude, it's more complex but I think it should be possible.
I might be able to try it later this week but I hope someone succeeds and posts an example before I do.