I have the same problem, I could not load font-awesome in my game.
The solution I've found is to simply
1. create a start menu state for my game.
2. In that start menu, I call the font I'm going to use. As expected it does not work.
3. When I get into the game state, it works.
Explanation: The browser only loads font-awesome on the first use. So you have to call it before using it. The solution was found in this article: https://hacks.mozilla.org/2016/06/webfont-preloading-for-html5-games/
Bonus tip: If you're a font awesome pro user the font is: "32px Font Awesome 5 Pro" instead of "32px FontAwesome", you can check in the css file search for @font-face, there is an entry called font-family, use the name displayed.