Would there ever be an advantage to using a 750CCA battery connected to a charger to power a mobile rig/amp vs using a 20amp regulated power supply?
I can't kick the thought out of my mind. Maybe I just get excited thinking about the trickle charger display changing.
It's a good option if you are in a location that loses power frequently.
You need to do it carefully, though. If you want it to work well, you need to have the right equipment and it needs to be set up correctly.
You would need to set the voltage output from the regulated power supply to whatever the manufacturer designated float voltage is for the battery. That will keep the battery topped off, but not over charge it. Ideally you want some isolation so the battery doesn't discharge back into the power supply, that can be done with a large diode, or a specifically built battery isolator. Using an automotive battery charger can be an issue as they are designed to quickly charge the battery. You'd want it to be a multi stage charger that will go back to a float charge once the battery is full. You'd also want to be careful with the amount of RF noise generated by the charger. Some of them can be pretty noisy.
If this is a car battery, you need to be really careful as they are designed for automotive use. Having them inside your home needs to be done carefully as they can outgas hydrogen when charging. Ideally you want it outside or at least in an enclosure with a vent outdoors.
Gel cells are a better option, if you have one. A bit less dangerous and designed for use indoors. They also make power supplies designed for communications use that have a battery revert option. These have a circuit that will keep a gel cell battery charged and power the radios. The battery is on the same buss as the radios, so when the power goes out, the battery is already on line. Those power supplies have better voltage control, regulation and RFI filters.