I don't recall if this has been mentioned before (but probably), but another (painstaking) approach is to make sure your book's ASIN match their actual Kindle edition. That way the download will find the right cover all on its own. Might not be *your* (i.e., Calibre's) cover, but it's better than nothing

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Calibre has always been able to handle that automatically if an ASIN is set properly via the Ids field on the metadata editor, IIRC.
Sidebar: I haven't used a Kindle in years. There may be a few settings in the conversion process that enforce setting a bogus ASIN, or no ASIN, or something that'll screw with this (thinking of the 'allow sharing' stuff or something).
(Sideloading on Kindle is a giant mess, Calibre manages to hide a lot of that from users, but it's a mess nonetheless).