Ok, now it’s getting clearer. So, this is what you need to do:
First, you need to register your own Bitbucket App for Auth0, you cannot rely on the Auth0 Dev Keys.
In Bitbucket, go to Settings > OAuth > OAuth consumers. There you create your application, afterwards you get a key and secret.
As for the Callback URL you register in this OAuth consumer in Bitbucket, it would be the one of your Auth0 tenant: https://YOUR_TENANT.auth0.com/login/callback
(so, not the callback URL of your Vue.js app obviously)
This key/secret, you use in your Bitbucket Social Connection configuration in the Auth0 Dashboard.
Put key and secret in these two input field where it’s marked red, don’t leave it blank / don’t use the Auth0 dev keys.
Now, the refresh token you got before, and this key/secret you got from bitbucket, you now use in the refresh token request https://bitbucket.org/site/oauth2/access_token
