There’s no inherent reason your pedaling has to actually power the device. You just want it to stay on as long as you are pedaling, right? So hook it up with a relay switch. The power just goes to turning the switch on, not actually powering device. You can use resistors to force a minimum amount of power before it works, if you want to make sure you’re not just barely turning them.
But, yeah, don’t connect directly to the PS2 to avoid corruption. It’s not designed to be powered off by the mains. Hook it to the TV. I’d also suggest not letting it turn itself back on. Just use a capacitor to keep it running for a bit when you stop. You don’t want to have the system constantly turning off and on in starts.