Generally speaking, ya that's how it works. Getting hacky you can try and whittle down LOD 1 to retain the same UV's but probably easier to just re-bake the diffuse outright.
Does game character must have one texture for all LODs or each LOD must have separate texture and different UV Layouts ? For example we have three LODs, as a result we need 3 different UV-mappings and 3 diffuse textures for each LOD ?