If you haven't already, I'd definitely start with inspecting the inside of the barrel closer for any signs of things like dirt or rust which you might be able to clean off relatively easily. Do the same with whatever power cable you're using as well. Does the issue persist with other cables? If so, and it's super clean looking, I wouldn't be surprised if it's an issue with the inner pin.
As you may have seen in your 1.4's teardown video, getting your hands on that component would require almost completely disassembling your camera:
https://youtu.be/Y_TFiA05tcc?si=FaktZb-LzhFXzr27&t=315If you're brave & believe qualified for the task, you can probably replace the barrel jack yourself. Maybe even upgrade it to one with locking threads like the 4K12 has too. Otherwise I'd highly recommend contacting support and sending it in for service.
One other, less elegant option might be possible (but I don't know of anyone who's tried it with a Chronos yet)- is maybe you could find a dummy EN-EL4 battery that works with it.