It seems like you're wanting something like OAuth instead of using Flask-Login. In case you don't know (quoted from Wikipedia), OAuth is a protocol that utilizes tokens in order to access resources on behalf of a resource owner. Think giving a user the ability to give out a valet key to certain portions of your site. Many sites, such as Google, Facebook, and Twitter use OAuth for authenticating third party clients in order to access certain user resources.
Right now, there's a split between the less flexible and less complex OAuth 1.0a and the more flexible but more complex OAuth 2.0. Many libraries exist for OAuth 1.0a in Python, but fewer for OAuth 2.0. However, there is a selection of those for OAuth 2.0 if stability isn't a top concern right now.
For the client, Flask-OAuth is available if you're going with OAuth 1.0a, and it is maintained by Armin, the Flask creator itself, so you can feel assured that it won't die. For the provider, there's an extension called Flask-OAuthProvider with OAuth 1.0a support. If you don't mind integrating it yourself and want 2.0 support, pyoauth2 provides you with both a client and a provider, though it looks less maintained.
Hopefully this helps you with exploring one possible avenue to utilize auth tokens, albeit without using Flask-Login. In my opinion, one shouldn't re-implement a protocol unless they understand it, so I recommend reading up about OAuth even if you decide not to use it. Many great articles exist on it, such as this article from Google and this one, too.