Multiple pulses from single button press
I deleted my other answer because I overlooked an important point in the part's truth table, which you correctly pointed out: while both count-up (pin 5) and count-down (pin 4) are low-to-high edge-triggered, the edge is ignored if the other count input is low.  

We'll suppose you have a decently debounced positive pulse, for instance obtained from the 555 debounce circuit you're referring to. Apply that pulse directly to pin 4 (count-down) and delayed by a short time to pin 5 (count-up). The delay can be created by a couple of gates. Then both inputs will see a rising edge, but for the count-down pin this will be while the count-up is still low, so it won't have any effect. The rising edge for the count-up input will occur while the count-down input is already high, so this will have an effect.  

So all you have to do is put pulse directly to the count-down input, and the delayed pulse to the count-up input.