After further consideration, I decided to replace my battery. I built a higher capacity pack (6.6 Ah as opposed to 4.5 Ah) using a PCM board that doesn't disconnect the battery near end of charge. The PCM top balances the cells at 14.4V, so, unfortunately, I had to replace my 14.2V charge controller, too. I went with the same model controller, custom programmed with a CV voltage of 14.4V.
My setup now functions as I had hoped. The load is now exposed to a maximum of 14.4V, so I can keep the panel and load connected at all times. This allows the controller to power the load from the panel when possible and from the battery when necessary.
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. This would maximize the amount of time I could transmit at full power (the KX2 will transmit at its max of 12W as long as input voltage is >= 13V). I don't know if a regulator like this is available to buy or feasible to build. Current draw while transmitting and powering a Raspberry Pi (for digital modes) is between 2.5A and 3A; I'm not sure if this would require a regulator with constant current...