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I am designing a website for a friend who is a confectioner, specialising in wedding cakes. I would like to have a "cake" preview for his potential customers where they can upload an image and see how it would look like on one of his cakes.

My object looks like this:

Image of a simple cake in Blender

However the "UV map" (print template) that my friend uses when creating the cakes looks something like this:

Two centrally centered circles

The inner Circle becomes the top layer and the outer circle is folded down around the cake to make up its side.

Is it possible to have a UV map generated that fits the print template/UV map my friend is using? Would I need to adjust that UV map vertex-by-vertex to fit that template? I have tried every option of UV unwrapping in blender that I could find, but so far everything that came out of it was far from what I would need.

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When you use correct topology everything gets simpler, faster and better.

1 Add a cylinder without caps

2 extrude one edge loop inward

3 press Ctrl F > Grid fill

4 Select the top external edgeloop and press Ctrl B to add a Bevel

Use the standard Unwrap function and scale edge loops as needed.

enter image description here

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For real world usage I think you'd use the Bevel modifier, limited by a vertex-group with only the top cap vertices, for a non-destructive workflow (meaning, you can easily edit the modifier settings later if you want to).
Then only during exporting is when the model will get triangulated.

I noticed that you have the bottom cap edgeloop as a UV seam.
With it like that, if you select all faces of the cake, then deselect the bottom cap faces, then hit U -> Unwrap, you should get almost exactly the UV cake template (the circle size might need some scaling to match).

Like in this GIF: Cake model step by step

PS you haven't said what you want to happen to the bottom cap, but I'm assuming it should also take the wall color. You can select the bottom cap faces and in the UV editor scale them down and place them within the outer circle.

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