I've discovered this is not an issue originating from Pi-Hole, but caused by a number of command line switches I've configured in Home Assistant, as detailed below.
But is there any way to alter the rate limits for API login? I understand they're probably in place for security purposes.
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.