Unfortunately my PSP-3004 battery pack died a long time ago, and I can't find any where I live. I tried using USB and original charger with the battery out, but none helped. How else can I power up a PSP?
My PSP-3001 works with the battery removed and the charger plugged in. It doesn't work when connected to my PC by a USB cable with the battery removed. I doubt your PSP-3004, the same model just a different region, would be designed any differently in this respect. There might be something else wrong with your PSP if you can't get it working with the charger. Maybe it's the charger that's broken.