As mentioned above, the problem seems to be the load order.
To ensure that jquery is loaded before various inline scripts get executed, we did the following:
Replace
(function($) { ... })(jQuery);
with
jQuery(document).ready(function() { ... });
in
…/wp-content/plugins/advanced-custom-fields/core/controllers/input.php line 102
…/wp-content/plugins/advanced-custom-fields/core/fields/_functions.php line 560
This solved our issues both regarding non-reponsive “Add image” buttons, as well as some conditional logic used to toggle field visibility.
Hope this may be of help to someone, and a big thanks to Elliot for a great plugin.