You mention that the "constant" wire is a heavier gauge wire than the "switched" wire. This is like my CD/radio. In this arrangement, all power to the radio including memory, clock, and sound come through that "constant" wire. The smaller gauge wire (switched) is simply a signal wire that allows the radio to turn on or turns it off when you interrupt power to this switched wire.
On my radio this full functionality, includes maintaining clock time, radio presets, what CD track you were on, last volume, and even remembering if the radio was on or off (If you go to battery "Off" with the radio on, when you turn battery back to 1, 2 or both the radio will come on by itself, just like in your car). I notice that there is even a small delay. Turn the battery selector switch to Off and the radio stays on for a second or two.
To make it work this way you connect the "constant" wire to either the Battery 1 or Battery 2 input posts on your battery selector switch (you can also connect to either the Battery 1 or Battery 2 positive terminals but I find it neater to do this wiring behind the electrical panel). Make sure to include the proper sized in-line fuse on this circuit.
Connect, the "switched" wire to the load side of a breaker within the panel. On my '84 E-28 I have a breaker labled "Radio" which goes to my VHF. I use the one marked "Tape Deck".
Drawback of this arrangement is that no matter whether you select Battery 1 or Battery 2 on your main selector switch, the radio will always power itself off only the battery you connected it to. Normally not a big deal with only occassional use unless you have a honking big stereo system. Also, when working on the electrical system, remember that even when you turn batteries to "Off" that line will still be energized.
I use a similar arrangement for my automatic bilge pump. I disconnected the "Stove" (I don't have Propane) and "Bilge Pump" breakers from the main bus behind the breaker panel. I wire to those breakers from the Battery 1 input post on the battery selector switch. Threfore the line side of those breakers are live no matter what the selector switch setting is.
Then I go from the "Stove" breaker load side to the "auto" lead on the pump. The manual lead is wired to the "Bilge Pump" breaker. So long as I leave the "Stove" breaker closed, even when I put the battery selector to "Off" I still have power to that "auto" lead. If I want to manually start the pump, I press the "Bilge Pump" breaker.
Other option is to wire both leads to the "Tape Deck" (or your equivalent). breaker on the panel. Advantage is that the radio will be powered off of whatever battery you have selected. Disadvantage is that it will go "dumb" every time you move the battery selector switch to "Off".
