How can I keep a steady voltage from car battery to power a switch even when battery voltage drops?
You should charge both batteries when either or at least the main engine is running.  Add a Schottky diode in series between each battery and the altenator output.  If you can, adjust the altenator output up a few 100 mV to compensate for the Schottky diode voltage drop.

Another possibility is to use a relay to short the batteries together once the main engine is running.  Deciding "running" isn't as easy as it may sound, but the advantage is that there won't be any voltage drop between the altenator and the charging batteries.

Once both batteries are being regularly charged, you shouldn't need separate power to cover the case when the back battery is dead.  Regularly deep-discharging a lead acid battery, especially one intended for a car, is not good.  That will ruin its capacity, and make it vulnerable to low temperatures.