Web Client is using http cookie and backend service is expecting claims in Bearer token.
Can Traefik transform incoming requests from WebClient's cookie and populate Authorization header with Bearer token (derived from cookie)?
Did you check Traefik plugins?
Alternatively you could create your own plugin in Go or (mis-)use ForwardAuth middleware with a tiny service (in any language), that returns status 200 and places the cookie in the header, which will be forwarded to the target service.