It will if your Auth provider (your server) follow standard oAuth2.
Also, it will work if you use this Auth provider to authenticate user to your Bubble app.
Actually, I understand that your user login to Bubble using Bubble auth. Doing this, you have no choice to manually create each step to authenticate them with your server.
If you want Bubble to handle refresh token,you must use user-agent auth and change your server to follow oAuth2 authorization_code process. OAuth 2.0 Authorization Code Grant Type