I did some testing and found the culprit - edit out this line of code in White Label CMS (line 26 in wlmcs-plugin.php) and now the two plugins happily co-exist (at least on my localhost they do):
include(ABSPATH . "wp-includes/pluggable.php");
One of the commenters on the wpMandrill support page posted the fix.
This line of code appears to be checking the existence of a native WP function,
wp_get_current_user(), and if it doesn't exist it includes the file that function is defined in.
From what I can see, wp-includes/pluggable.php is a core file, and should get loaded by wordpress itself, so I'm not sure why WLMCS needs to check for it. It might be better to use an action hook like 'plugins_loaded' to accomplish whatever goal is desired.