Perhaps the clock's electronics or electro-mechanics is sensitive to voltage/current, which is notably higher in alkalines?
Does the clock contain any (white) LED? I have a few simplest torches with LEDs connected to batteries directly, using no voltage/current limiting electronics, and if alkalines are used there LEDs get damaged with too high current (zinc carbon's voltage tends to fall very quickly on any substantial load). I can safely use "weakened" alkalines there though - the ones that stop being usable in MP3 player, remote control, automatic blood pressure meter, and the like.