The first page of an ePub is the cover, that is the way that ePubs work.
Most non-cover covers end up with a small image size. So, one work around may be to search for a cover that is smaller than (say) 20 KB. This is the default size for the "cover" option of the Quality Check plugin, which I recommend.
When you generate or select a new cover the cover.jpg gets replaced. To get this into the actual epub you need to do an ePub to ePub conversion.
If an existing image is selected as the cover, it will still be in the ebook if you replace the cover.
The separate cover is only written to disk if that is what you specified in Properties -> Savng books to disk.