In a similar report to yours, where an iPad refuses to charge (after full discharge) with the Apple OEM Power Adapter and cable - the contributor reports having been able to recover the iPad to a working state by using a third-party USB cable.
From a technical perspective, whilst you’d expect one USB cable to be much the same as another, there are some differences other than the gauge and quality of the wiring. USB cables usually incorporate additional passive components (resistors) that, when the cable is connected, signals certain conditions to the host device.
As such, if the related report is accurate, changing the OEM cable for a third-party cable may have a positive effect. After recovering charge into the iPad, the original OEM cable was reported to be working as expected. Whilst you may not be able to reach an Apple Store, USB cables are commonly available.