Yes
Source: Experience
...
No, really.
With the help of Pokemon Home, leftover Pokemon from completing the Dex the first time, and a brother with Pokemon Scarlet, completing the Pokedex a second time was actually super easy. So I transferred a dex worth of Pokemon to a new save account (I started transferring after I got the dex on the new file), did a bunch of story stuff, and eventually found my answer.
The first thing I tried was talking to Jacq after the cutscene when you first meet Geeta, but to my disappointment, Jacq wasn't present in the classroom.
My second attempt was after the announcement of the Treasure Hunt. I was able to go back in the school and go to the classroom, but when I talked to Jacq, he wouldn't initiate the dialog where he checks your progress on the Pokedex. I even tried going into the Pokedex and getting the achievements from the Pokedex so I could get the message to see Jacq for a reward (realizing now that I should've taken screenshots of exactly what Jacq and the Pokedex were saying but too late now), but even after that, the dialog wouldn't trigger.
My third attempt was after I got the ability to ride Miraidon. I went back to the school and went to the Biology Lab, talked to Jacq, and the dialog I needed triggered, so I got the shiny charm. Now I'm not sure if getting the achievements from the Pokedex is necessary, because I wasn't about to sit through getting 40 achievements in a row for a second time, but anyways...
TLDR: You can get the Shiny Charm as soon as you have the ability to ride Koraidon/Miraidon. Go to the Biology Lab in the school, talk to Jacq, and he will give you the Shiny Charm.
I also have pictures, which hopefully, you can see them.
Now of course, I could be totally lying (I'm not), so hopefully, the fact that the background is the same for both pictures is a sure sign I got the shiny charm without getting any badges.
Hope I helped!