How do you apply timezone formatting to a date?
The arguments you pass to the function are wrong; in particular, what you pass as last argument should be the one before it.
The arguments the function accepts are format_date($timestamp, $type = 'medium', $format = '', $timezone = NULL, $langcode = NULL), but your fourth argument is an empty string, and then you pass the timezone identifier as last argument, which should be the language identifier.
You are also using '$date' as first argument.

Try with the following call:

format_date($date, 'custom', 'D d/m/Y - g:ia', date_default_timezone_get()).