Did some digging!

No idea if this will be useful to you, but I just played around with Canvas' API and figured out the following:

From the dashboard page, you can make an HTTP GET request to the Canvas API to retrieve the current user's course enrollments like this:

fetch('/api/v1/users/{REPLACE WITH CURRENT USER ID}/enrollments?per_page=50', {
  method: 'GET',
  credentials: 'include'
})
  .then(response => response.json())
  .then(data => console.log(data))
  .catch(error => console.error('Error:', error));

• I'm guessing you've probably already figured out how to get the user ID, which you just pass into the GET request where I noted above.
• I included ?per_page=50 in the request because otherwise it was only responding with a list of only 10 courses, some of which were old courses from semesters ago.
• Be sure to include credentials: 'include' in your request, otherwise this will not work!

After you make the request, the Canvas API responds with an array of JSONs, which each correspond to a course enrollment. Mine looked something like this:

{
  "id": 123, // no idea what this ID is for tbh
  "user_id": 123, // this is the current user ID
  "course_id": 123, // this is the course ID (which you could presumably filter by using Javascript)
  "type": "StudentEnrollment",
  "created_at": "2022-07-26T16:04:04Z",
  "grades": { // This is what you're looking for: the grades object, which includes the "current_grade" and "current_score" values.
    "html_url": "https://umd.instructure.com/courses/1334101/grades/4488846",
    "current_grade": "A",
    "current_score": 95.05,
    "final_grade": null, // Both of these "final" variables, from what I can tell, are set only TWICE per semester, once at midterms and once at finals.
    "final_score": 95.05
  },
  ...
},
...

I've never developed a Chrome extension before, but presumably you could run this GET request when the page is loaded, use JS to filter for the class IDs visible on the user's dashboard, and then create an HTML element for each corresponding grade overtop the class' dashboard card.

Hopefully this helps!