Turning the ignition to "on" will run the fan (if on) and power the ECU and dash lights. In comparison to the radio, the only significant drain in that is the fan, which you can simply switch off.
Alternately, for less work, you can connect a switch which will power on the radio without powering on the rest of the things in the car. You need a $1.99 DPST switch. In the "off" position, it will connect the switched power in the radio to the accessory power in the car. In the "on" position it will connect the switched power in the radio to a constant +12V source.
Alternately, you'll need to disconnect the keysense for the ignition switch. The speaker for the door chime is in (I believe) the #1 integration relay, so to remove it, you'd need to open the box and de-solder it. I wanted to tap into it for something else at one point, but decided against it. (The headlight auto-off doesn't work with my remote car starter, since the car keeps running until I get out and shut the doors, so I wanted to use it to warn if I leave the lights on. It ended up being easier to use the alarm's "armed" output to fool the car into thinking that I opened a door for 1.5 seconds, which activated the auto-off.)