This problem looks similar to an Issue report on the Mapbox GL JS GitHub repository, which was also experienced by two users of our service recently, on desktop Google Chrome.
One piece of information missing from this question is whether this 403 response was cached by the browser.
In the case that it was, it aligns with the issue I linked above. Clearing the Chrome browser cache solved it for our users and the reporter of the GitHub Issue, but this had to be done in a specific way.
Methods that worked, in Chrome & derivatives:
Other cache clearing methods did not work, such as Application -> Clear site data or a Hard Refresh. I don't know why.
I suspect the issue might have been caused by the usage of an old v1.x Maplibre/Mapbox GL JS version in combination with a years-old service worker cache of Mapbox tiles.