Seth's comment about the remote xserver sessions led me in the right direction. VirtualBox can be directly started on its own X session. In the example below, we start an Ubuntu 15.04 image.
From the command line
Create a file named startx_ubuntu1504
(or whatever you want to call it) in your home directory with these contents:
virtualbox --startvm Ubuntu_15.04 --fullscreen
You can then run startx $HOME/startx_ubuntu1504 -- :1
to start the VM.
From the login manager
Create a file named /usr/share/xsessions/vm_ubuntu1504.session
with these contents:
[Desktop Entry]
Type=Application
Exec=virtualbox --startvm Ubuntu_15.04 --fullscreen
Name=VM Ubuntu 15.04
Comment=Run VirtualBox image in its own X session
You should then be able to select the X session from your login manager. For example, I'm currently running Xubuntu 2016.10 with the lightdm login manager, and my current login theme has a session dropdown near the upper right corner. This lets me select between any number of virtual machines or my non-virtual desktop. Very cool!