Ok, I think you're correct in that the bot should have that user right already.

Alright, I tested by removing the $wgFlaggedAutoReview = true from my LocalSettings.php and the reverts are still showing as automatically reviewed. I actually do not think we need this field to be set. (default 3 seems to be ok!)
I think we just needed to assign the autoreview user group to the AutoModerator user. Should this be a configurable option per wiki or hardcoded? Is there any reason a wiki would want AutoModerator's reverts to show up as manually reviewable?

$wgGroupPermissions['sysop']['stablesettings'] = true;
$wgFlaggedRevsRestrictionLevels = [ 'autoconfirmed', 'autoreview', 'sysop' ];
$wgFlaggedRevsProtection = true;
$wgGroupPermissions['AutoModerator']['autoreview'] = true;

Sam, great questions.

After an investigation into testing configuration locally I have uncovered the following configuration to suffice: