Yes. You have some control by including/excluding certain sites from the rules, i.e. you could exclude common shopping sites from the CC rules, and we imagine most folks trying to stop (accidental) exfil through AI chatbots are going to block most common providers there.

Warning (instead of blocking) is possible. Due to the security logging (plus, if wanted, SIEM integration) you get the flagging.

Can you run data through a local or corporately-situated LLM instructed to "Assess whether this looks potentially sensitive" via a given set of rules, and flag it to the user or a supervisor to proceed/abort?

Not in the first version, but this is an area we are working in.

what kind of hooks and feedback/UI interaction is available for third party integrations?

We support the standard SDK for this which is also used by Chromium based products:

https://hacks.mozilla.org/2025/03/improving-firefox-stabilit...

That's how Trellix and Broadcom integrate with it as well.