I have now got acceptable performance. Newrelic showed that admin-ajax.php was triggering a bug in BuddyPress Docs that was adding 3.5 s to each response. The plugin author is addressing the issue.
Inspection of Firebug log shows that an AJAX request is made on each keyUp. As Henry’s link suggests and my Newrelic logs show admin-ajax is consuming most of my sites resources. Developers need to reduce its use.
For small sites like mine with a perhaps 100 members in total the entire directory could be pre-loaded when the Compose tab is loaded. This would reduce server load and give 100ms response to the user.
How can I flag this as a feature request to core developers?