I'm thinking that you need to replace the logic board in that iPod, cause it sounds like its the dock connector causing the issue, and the dock connector can't be replaced easily w/o a motherboard swap.
I was able to mod my iPod 5.5 Gen with the 3000 mah battery I found on ebay too, I got from full to flat in about 2 days of continuous music playback (I own a 24 hour store with music) with the backlight on. The battery indicator is more accurate with the mod than without it, with the old sh!tty battery, the battery indicator would say empty for like an hour before it died. To fully charge my iPod, it took about 5 hours but your charging time may vary since I used a high amp charger. The battery indicator will become accurate after a couple of battery cycles.
If you can open up your device, try reseating the display data and power cables, and if that doesn’t work, try contacting an Apple Certified Retailer, you may be eligible for a free screen replacement.