AL8860 LED driver does not output required constant current
From your first experiment using a benchtop PSU in CC mode, the output voltage reaches nearly 12V if you target is 1A. It means that Vf for your three LEDs is about 12V (4 V per LED), which sounds reasonable. On other words, to get 1 A through the LEDs you need to have a source in excess of 12 V. 

However, the AL8860 is a switching regulator, it can't output 12V if Vin = 12V, it must have some room, because it has some limit on duty cycle (75% maximum recommended), and will stop working if Vout approaches Vin. Try to apply 14-16V as your input, everything should be fine then.

From AL8860 datasheet, page 7, to get 1A LED current at 10-11V level, you need at least 14 V of input voltage: