Export saves your whole image however it is currently cropped, and does not pay attention to your viewing zoom. This is normal for all image editing programs.
You can, easily and efficiently, have more than one version of a given photo showing in LR: by making a "virtual copy". This inherits all the same current editing, but can be further edited independently, or independently cropped to show whatever smaller part of the photo you want. The full photo still remains in place too, alongside. Then you can select and Export this new, specially cropped virtual image version - saving out an external file which only includes that smaller part of the photo. You can choose whether to resize this Export (to a certain number of pixels): if you un-check this resizing, LR saves out the original pixels as viewed inside LR at "1:1 scale".
Once you are finished with this (extra) virtual copy version you can safely delete it - doing this does not affect the master version, nor the source file on disk.