GlobalNavButtons uses Mustache to preprocess your global configuration, which its syntax is very similar to wikitext. Unfortunately, that includes its sanitization rules, so you can't inject an image tag in there.

You'll need to fork/modify the script and include support for an image or icon parameter, whether it's from user input or from Fandom's global-explore-navigation__icon, starting from here:

https://dev.fandom.com/wiki/MediaWiki:GlobalNavButtons.js#L-77

Edit: I utterly misread your post, apologies. I'll need to create a new one below

With how the script injects icons, it must be something that global-explore-navigation__icon accepts. I do not know how the icon part is handled after the Mustache expansion and appending to the DOM, you'll need to ask the script's maintainer(s), or wait a bit until someone else responds to this