WordPress with phpMyAdmin - 404's everywhere

WordPress shouldn't be interfering with phpMyAdmin, since the plugin loads it in a isolated iframe.

As one of his specifications for the project he wants ONLY WordPress installed on his server...

The plugin is, nonetheless, still phpMyAdmin (albeit 'wrapped' in the WordPress UI). In other words, you've already installed it ;)

...to avoid the hassle of updating and maintaining other software...

'Software' can be a dangerous term when talking web-apps - that's not to say don't use it at all, but for some it can conjure up thoughts of blue screens and runtime errors ;)

In other words, just stress that PMA is simply a collection of files on the server - it has no database of it's own, it's effectively stateless, and removal is as simple as RMD /phpmyadmin.

...he wants to be able to make all necessary administrative changes from the WordPress Dashboard

Despite the points I've already made, if it is absolutely essential that there is database management access within the dashboard, I'm about to write a quick alternative that uses phpMiniAdmin instead (that's how I stumbled on this question oddly!), and I'd be happy to share it for you to try out.

As @molnarm pointed out in the comments, why not just removed phpMyAdmin and connect to MySQL over SSH, using something like MySQL Workbench or Sequel Pro.

You would have a much easier and faster way to interact with MySQL and could delete the nightmare that is phpMyAdmin.