I was able resolve the issue where the overlay wasn't showing up however it isn't inserting the shortcode into the text area.
In a case like this you will need to manually add the shortcode as a workaround to this. For the Types shortcodes in Divi it would be {!{types field='my-field'}!}{!{/types}!}
So you will replace my-field with the slug of your field.
Please let me know if this workaround is suitable for you.
The field and views button will allow you to insert this shortcode. Essentially the end result will be the same just that you're going to add the shortcode to the template manually.
I'm able to identify partially the issue and it has to do with the css z-index for the shortcodes popup but even when I adjust the z-index the shortcode still doesn't get added to the editor.
As per your video I get the impression that you want to insert the shortcodes for the custom fields into your template. Also will you be making use of the gutenburg editor as well ?
It's a bit messy because we have to write or copy and paste that shortcode for a lot of fields and we have to remember every internal name. Even if it works, it's a temporary solution in a startup phase.
Not sure what can be done about this, however i've escalated the thread to our 2nd tier supporters with the issue replicated on a sandbox site so that they can do further investigations on this.
I will update you once I get a response from them.