I don't think it's a faulty battery problem, but it is certainly a possibility. The reason is that it does the same (screen lit for a few seconds, flickers and turn off) whether the charger is connected or not. So I think that the current from the charger goes through the battery even at full charge and the battery has a limit on the max load it supports. The battery for the Tab 3 most probably is not prepared to supply as much power (W) as the Nexus 10 requires.
However, if you want to give it a try you're very welcome. The part number is T4500E and the ones I found cost 13$ on eBay here on Europe. Wouldn't be surprised if you can find them even cheaper in the US.
Your symptom sounds like a bad battery unable to maintain voltage above 3.6V. When voltage drops to 3.55V or so, the tablet will flicker and turn off like that. The USB charging port does not supply enough power to keep the tablet powered up. You need the POGO cable with a 2A charger to keep the tablet on with a bad battery. If your battery is not bad, it should be able to keep the tablet on, as the tablet current it draws varies depending on activity, and will be under 1A with moderate activity. One thing you can do is try to go to the recovery screen and see if it stays on then. If it also flickers and powers off in recovery, I'm pretty sure it's a bad battery, as recovery screen won't be drawing much current. While I'm curious whether another battery will work, I already have 2 working batteries now, so I don't think I'll perform that experiment now