I see that you asked about converting and sideloading to your Kindle, and therefore the above answer is your best bet.
In case you are interested, I have been playing with converting epubs and emailing them to my Kindle, and have found that the only way to do this is to convert to "old" MOBI filetype. Through trial and error, I can verify that Amazon will not send the "new" MOBI format, nor AZW3.
Kovid confirms this here:
https://www.mobileread.com/forums/sho...16&postcount=3