You won’t be able to do it in the native Internet / Silk browser that comes with the device, so you’ll need a different way to get the Fully Kiosk APK.
Is the ‘Downloader’ app from the YouTube video or the written instructions still available?
I saw a thread on Reddit where someone mentioned that Amazon blocked the Downloader app from Show 15s, but since I already have it installed, I wasn’t able to verify that.
I was just poking about the X-plore File Manager, which was also mentioned in the written steps, and it has some options for getting files between devices. For example, I downloaded X-plore File Manager on my Android Phone, enabled the ‘Wifi Server’ on my phone, then back on the Show 15 I was able to access the ‘Wifi Server’ category from the right-hand panel to access the files in my phone where I could long-press the select button on my remote to bring up a menu to copy, then navigate over to the left hand side into my Download folder and paste the file.
X-plore is pretty clunky with a remote and I had to watch an ad to enable the copy functionality, but it could be an option. (Note that I’m generally hesitant about apps having full access to my file system, so I uninstalled it from my phone when I was done).