Which mice are compatible with PS2 games that allow for this, e.g. Dirge of Cerberus: FF7? Or, what might cause incompatibility, or help diagnose this? A Logitech G5, plugged in directly, was no-go with that game.
Maybe you are experiencing some incompatibilities because of the hardware glitch that the PlayStation 2 USB OHCI controller has. The documentation states that the hardware has problems with 64-byte transfers, if the buffers are not aligned to a 4-byte boundary. It says that USB devices that truly conform to USB 1.1 will be compatibe with the workaround it offers, but some devices aren't truly USB compliant. So you may want to just try a different device.
What sp193 said, also one thing I discovered, or at least on my end was that you can't use a wireless mouse and keyboard setup on the PS2. I had to use a separate mouse for mouse control for those games.
