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You don't say what kind of a radio you have but I'm assuming it's a modern solid-state one-- if the voltage drops below a specific level (probably around nine volts) the unit is probably shutting itself off. This coupled with the fact that the starter won't crank after a shutdown episode signals a classic low-voltage problem.

Do you still have the dual battery setup? What type and how old?

Your alternator really isn't intended to bring the battery(s) back from a discharged state. When they are really low, the alternator will overheat trying to bring them back. If your batteries are only a year or two old, I would take them out of the car and charge them for a couple days using a battery charger on the 2A setting. You might also give them a test using a "load tester" which will actually put all the current through a heating element (looks/smells like an electric toaster) and allow you to measure the battery's strength. Your FLAPS may have such a tester available.

Anyway, once the batteries are topped up to 12V, you are then ready to test the charging system. When you turn the ignition "on" what does the idiot light do? It should glow brightly. When you start the car and rev the engine past about 1000 rpm the idiot light should go all the way dark. If there's ANY deviation from those results, e.g., no idiot light on with key on, or a dimly glowing light at idle rpms, this is indicative of a fault with the charging system.

Last, with the engine running, measure the voltage at the battery terminals with a voltmeter. It should read around 13.5-13.8 volts. The alternator puts out 14V when it is working properly, but there's a voltage drop going from the alternator to the starter post up the big wire and to the batteries. If it reads less than that, or reads the same whether the engine is on or off, there's your problem.

Do NOT disconnect the batteries with the engine running as this will result in the immediate destruction of the alternator.

Good luck!