When looking in the type certificate for the different configurations of the A320, I found a lot (really, more than 50) weight variants listed. But none seemed to mention the actual engine paired with that specific weight variant. Is there something I am missing, or is there another way of finding out?
Each section will list the specific models that it applies to. For example, in the Maximum Certified Masses section for the A320 (1.III.13), you'll see the first section applies to the A320-211/A320-212/A320-231, and it lists the 19 variants of these models.
Next you consult the engines section (1.III.5), which will tell you, for example: A320-211 - Two CFMI CFM 56-5A1 jet engines (MOD 20141), or CFM 56-5A1/F jet engines (MOD 23755).