I did some more digging at the issue. I managed to duplicate the Your Kindle App needs an update to download using Kindle for PC 1.26.
This required a 2 step process.
After opening Regedit, I went into HKCU/Software/Amazon/Kindle/User Settings and changed the value of isKRFDRendererSupported from true to false. At this point, it looks fine but as soon as you open Kindle for PC 1.26, the value gets changed back to true.
To prevent this from happening, again in Regedit, right click on User Settings and then Permissions. In Permissions, click on Disable Inheritance. This will ask if you want to do with current inherited permissions. Click on the Convert Inherited Permissions into explicit permissions on this object. Once that is done, step though the listed users, double click on any that don't show Read as the Access and remove the check mark from Full Control leaving the one on Read.
Perhaps not oddly this seems to block all KFX downloads for K4PC 1.26 and 1.28.
Edit: Also checked with Kindle for PC 1.29.58059 and still did not get KFX files though I did re-enable Full Control for System, my user and Administrators before the install and removed Full Control after the install. Without that, the install seemed to work but Kindle for PC came up still saying it was 1.28.55076.