Your steps 1 thru 6. I suggest that you edit those, in order to be more specific - in details - for each occasion that you mention “account” and “accounts”.
I know, that you know what you are referring to, but I do not know what you are referring to . . . and though I may guess well, you would help both your cause and Brave Support by not leaving any doubts.
Log into any Instagram account that has three linked accounts. Those linked accounts are on the Internet, and they are each linked to the Instagram account by means of Instagram account settings.
Save the data and passwords of each of the three linked accounts.
Close Brave Browser and wait for some time, and then Restart the computer.
Now Open Brave Browser and try to switch among the three linked accounts ← that will show a loading icon but either Stop there . . .
Or that creates a Pop-up where the stored data is filled, and I need to click on a Login button.
In step 4, . . . actually, I do not understand your step 6. You wrote “alternate process” ← What alternate process?