This is probably not possible, but let us see if there are more fluent users of Data Transfer modifier here.
Goal: transfer the custom normal from this continuous piece onto these 2 split pieces
You might not see it but the difference is normals on split pieces are not smooth like the continuous piece:
I have some experience with Data Transfer modifier, but for this one, strangely, I couldn't get it to work.
Alternatively, if it's possible for me to set a split normal to point to 3d cursor, then I could have fix it without transferring too, but I wasn't able to do that without crashing Blender.
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I am facing a low-to-low, split vertex normal transfer.
Don't know if it handles discontiguous normals though.
Basically I need to use "Projected Face Interpolated" option here and set both "Max Distance" and "Ray Radius" to a small but workable value (just like when you bake textures). To limit affected vertices I use "Vertex Group" to further limit the transfer to only vertices with normal mismatches.
https://docs.blender.org/manual/en/dev/modeling/modifiers/modify/data_transfer.html
The result looks pretty good: