The radio is not at fault here, get that idea out of your head.

This all reads like a battery that has a fractured connection between a battery post and internal plate or bad battery cable clamps or something similar. I sure could be wrong, but I've had this happen before - and I found it by turning the headlights on with the engine not running - and using pliers to wiggle the battery terminals/posts, when I put stress on the one post, the lights cut in and out, which indicated a bad connection inside the battery.

A new battery fixed that.