How to fix "Charging not supported by this accessory" problem?

I'm trying to get a used iPhone 3GS to charge properly but continually get the "Charging is not supported by this accessory" message. The phone is running iOS 6.1.6. The seller said it had been jail-broken at one point but that that had been undone (restored via DFU mode).

I've tried three cables: an Amazon branded one, a Belkin and an Apple (from an iPhone 1). All gave me the "not supported" message. However, all work fine with my other iPhone 3GS (never jail-broken).

These are the other things I've tried but have proved fruitless:

- Restoring the iPhone,

- using an Apple USB wall charger (directly, not through a cradle) with all three of the above cables

- Installing two different dock assemblies

- Installing a new battery bought from ifixit

- shorting the data lines of the cable (saw it on Youtube) on the phone side of the cable

Interestingly, cooling the phone either in the fridge or on an ice-pack before charging seems to increase the chances of a successful charge, but I worry about the condensation that builds up on the phone as it warms up.

Sometimes the phone trickle-charges despite the error message but other times it does not. If it does, it usually takes overnight to do so. Sometimes if I jam the cable hard or stick it in partially or wonky, it’ll charge ok without the error message for maybe 20 seconds. Rarely will it charge all the way without the "charging not supported" message.

At this point, I can only conclude that there's either there's a motherboard hardware issue or that Apple's tight-fisted attempts to destroy functionality of 3rd party accessories that somehow don't meet its hardware standards via software/firmware updates has essentially bricked this phone.

Any suggestions for further troubleshooting would be appreciated.