Line 457 in SearchEngine.php is:

return $wgServer . wfScript( 'api' ) . '?action=opensearch&search={searchTerms}&namespace={namespaces}&suggest';

Looks like the same error is thrown by the web site, not only when running update.php.

Adding this line to LocalSettings.php fixes the problem, but should it be necessary?

require_once( "includes/GlobalFunctions.php" );

It's a change in the order of the AutoLoader running, with respect to the settings also.

I'm not sure if this was was "supposed" to work, or was incidental

I believe it was an accident that it worked. Moving it to $wgExtensionFunctions should do the trick.

Never try to call MediaWiki active code from LocalSettings.php main area -- *all* MW functions and classes should be assumed unsafe to use.

If you need to call into MediaWiki *after* basic platform initialization, use $wgExtensionFunctions to add a callback that runs at that time.

Brion: even static methods are considered unsafe? They are by nature designed to be called without initialization from their class.

The classes aren't in scope (ie haven't been loaded) so aren't accessible

Thanks. After trial and error, I rewrote our code to set $wgMWSuggestTemplate via the hook "SetupAfterCache". This seems to be the only hook early enough to change $wgMWSuggestTemplate in time.