I think you would want to do a GET request to the /people endpoint.
For example, we use emails as usernames, so if I hit the /email@example.com endpoint, the id field in the response contains my id (id: 1234).
Then I hit the /members/people/1234 endpoint to return my membership.
Hope that helps?
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There is a "magic" call :-) that covers this use case: GET /api/core/v3/people/@me returns the person object for the authenticated user.