Slack now connects once, from your organization settings, rather than client by client. An admin installs it there and sees every client's channels on one page. Each client then has its own Slack page in Settings where you link channels, pick the default, and mark whether a channel is just for your team or one the client reads too.
That label matters. Ask the agent in chat to post something and it goes to the client's internal channel unless you say it's for the client. Before anything lands in a channel the client reads, you see the whole message with a Send and a Cancel button, and nothing goes out until you press Send. When someone mentions the agent in a client-facing channel, it leaves your internal Brain notes out of its answer and sticks to the client's own results.
The client home page also picks up a card for the week in that client's Slack. It sums up what the conversations were about and lists the threads worth opening, each with who's in it and a link straight into Slack, along with any to-dos those conversations produced. Only your team sees it, and it stays out of the way for a client with no Slack or a quiet week.