Admin Tools requested to run the Wizard again, which regretfully and stupidly, I did. Yes, I know I shouldn't have after the fact, but what's done is done.
After running the Wizard, the connection to the site (administrator & public) would hang indefinitely and the connections to the database would bloat. The Joomla default .htaccess configuration was active. JCH Optimized was temporarily disabled, and the cache cleared. The only resolution was to manually disable Admin Tools via the terminal. With Admin Tools disabled, the site performs optimally.
Even after subsequent reinstallations, Admin Tools doesn't seem to be working well with RDS and the site hangs when trying to save settings from within the component itself.
Akeeba Backup has similar problems. It will not proceed past the "Backup Database" step. Since AWS provides regular backups of our RDS database, I can simply change the Akeeba Backup configuration to backup site files and skip the database, however, this doesn't address the problem.
I suspect it may be a database connectivity problem between Akeeba and RDS, but need some advice as to what parameters I should be investigating. The site itself works fine under RDS.