"I would still like to put a voltage regulator between the controller and the load, no longer to protect the load, but to keep it powered at >= 13V as long as possible when running on battery."
Ain't gonna happen. You'e got something grossly wrong in the setup, and I think you are also under some misimpressions about how it can all work.
First of all, you will never have the battery powering the radio at more than 12.8 volts, which is about the maximum you'd eer see on a lead acid battery of any type. (Lithium might go higher, but that's a whole other kettle of worms.)
Anything over 12.8 volts (typically 12.6) on a lead acid battery is a float charge, which disappears after just a few minutes under any load. So, your battery is not going to give your radio anything above 12.6-12.8 volts.
Second, putting a regulator after the controller is totally counterproductive. The controller is supposed to have a regulator built into it. Either it works, so no second regulator is needed. Or it is broken--and it needs to be fixed, not kludged with a second regulator.
What you might want to do is replace the charge controller with an "MPPT" type controller. They can be 15% more efficient than pure DC simple controllers. And they may gain another 15% because they can take the low voltage in the early/late hours, or if the panels are partly shaded, and still boost it to effective charging voltage.
From what the OP is posting, there's just too much wrong, it needs to be redone right, not kludged on the kludges.