Currently, it seems like there's no "proper" way to manage Acquia Connector configuration (using the core configuration management system).
Generally, the D8 core configuration management workflow is to set up a site locally, run drush cex to export a complete configuration set, and then import that in a remote (Cloud) environment.
The problem is that Acquia Connector uses the configuration system to store volatile data that is really more closely aligned with "state". Quite often, this data will change on the server, so the next time you import configuration, you are actually importing stale configuration. This problem is being discussed at
#2864697: Move subscription and SPI data out of config and into state API β
Theoretically, you could work around that by using a tool like Config Ignore. Config Ignore would allow you to tell the D8 CM system to simply ignore Acquia Connector configuration by excluding acquia_connector.settings from configuration imports and exports.
The problem with this workaround is that it doesn't work for the initial site install. If you have a configuration set with Acquia Connector enabled and its configuration ignored by Config Ignore, what seems to happen is that the site is installed, Acquia Connector is enabled, but it remains unconfigured.
I'm not sure if its configuration is getting created when the module is enabled and then deleted by Config Ignore, or if the configuration is never created in the first place and then Acquia Connector's install hooks aren't run to populate it. Either way, it's broken, and I can't see how it's supposed to work :)