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Wild idea (NOT TESTED):
This would not be for booting a whole second kernel, so it might fail badly if the distros were on wildly different kernel levels. For a graphical environment for X11, try [Xephyr](https://www.freedesktop.org/wiki/Software/Xephyr/) which makes a second, inner X server display. In the host graphical environment, you would install and run xephyr, and learn its display ID, for example, `:1`.

In a terminal still in the host, chroot to the root partition of the guest, then set the display environment variable, and then switch to the correct user and try running graphical programs. For a desktop environment like Xfce4 those start with something like `xfce4-session`.