Any particular reason not to leave it on all the time? That sounds simpler. It's not like the amp will consume much more power than the stock radio to drive about the same volume. Power is power, if you only play at 2W, it'll use 2W whether the amp does it or the radio does it. The radio's not also going to spend another 2W to "drive" the amp through the speaker output, it's going to be milliwatts.
Personally I just ran an aux jack to RCA to my amps in the back and just use that all the time. It's not like I have to run it at full blast all the time, if I want it quiet it remains quiet. But it still sounds better than the stock radio.