After browsing through the HP support forums, I have found that the problem is that the BIOS disables the onboard video when a PCI-Express video card is installed. There is no setting in my BIOS for "SurroundView", which supposedly the DC5xxx series has.
Therefore, the solution seems to be either to replace the Radeon X300SE with a card which supports dual monitors, or remove the Radeon and find a PCI video card, which either may or may not cause the onboard video to be disabled. This is further complicated by the fact that the small form factor machine uses small form factor cards.
Also, FYI, the Radeon X300 SE put in by HP is a modified version for the small form factor box. It has only two outputs - DVI and S-Video. While the standard X300 also has an analog VGA output, and is therefore capable of doing dual-head operation itself, the version installed by HP is not.
I will post again when I have found a solution.