You could temporarily increase the session limit, e.g.
sudo pihole-FTL --config webserver.api.max_sessions 17
But this should really be addressed by the third-party maintainers.
They have to explicitly close the API session after finishing their call or calls, which is different to v5, which didn't have a REST API with session based login.
For several calls in a row, they should collate several API calls to use the same session.
They would need to do so on successful calls as well as on ones that would encounter exceptions, e.g. if the connection dropped temporarily during communication.