Thank you for the shared pictures.
Please, allow me to start by saying that changing the way the font is registered will not be sufficient. There is a known limitation regarding the usage of external fonts during design-time. According to this article, under the Automatic font embedding section: "During design time, there is no support for searching an application's use of fonts and automatically embedding the fonts in the application's resources.".
With this in mind, in order to avoid the designer errors, you have to install the font on your machine, if it is not already installed.
Additionally, you can register the font in the static constructor of your window. Please, find a sample project, where I have setup the Font Awesome 5 Free Regular font. Note, that on my end I had to run the project and rebuild the solution several times for the Visual Studio designer to refresh properly and display the glyph in the designer(removing the errors).
I hope you find this information helpful.
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