I am afraid Pagure's API doesn't support (at least for anybody who is not admin) operations with keys (including reading its settings). What we can do is making a simple request (with the current key) and based on the response we can determine (or estimate?) whether the key is invalid or valid. Of course, even with a valid key, it is not assured you will successfully perform the required operation (ACLs could be wrong).
The check can be done before the operation (request new repo or branch, do a fork). In case of negative response, we can show a hint to a user on how to get a valid new key. The same hint can be shown in the case of no key present in the configuration.

That makes sense logistically. In that case, the solution to this will be making a simple request with the current key and estimating the validity of the current key based on the response, as you said. Thank you very much for your analysis of the Pagure API capabilities.