Use Execute Async for Custom AI Permissions with Flow Triggers

What does this MR do and why?

Use execute async for custom AI permissions with flow triggers.

This is behind a feature flag.

Issue: Role-based permissions DAP - Background execute... (#578555)

References

This was introduced in Add concern for role-based permission checks (!213979 - merged) and Check Customizable Permissions for trigger_ai_f... (!214325 - merged), but we need to change which permission we check.

This also renames the methods in the concern in response to feedback here: !214325 (comment 2940629163)

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