Display Suite: Edit tab missing on user profile
Further answers are welcome, however, I eventually found an easy solution to this issue. Because Display Suite supports tokens, I was able to create a custom field that outputted an edit button. Here's an example of the html I used: &lt;a href="[user:edit-url]"&gt;&lt;i&gt;CUSTOM ICON&lt;/i&gt;&lt;/a&gt;. 

In order to hide this icon from unauthenticated users, I enabled field permissions which is available when the Display Suite Extras module is enabled. Once enabled, you can adjust what fields are displayed for each user role in the permissions menu. Hopefully this is helpful to anyone experiencing a similar issue.