Can I change my billing date within the following months billing cycle?

Yes, you can make this change.

Illustrative example below: 

Luvo's current billing date is the 21st of March. 

Luvo updates his billing date on 23rd March. 
Luvo's new billing date is now the 15th 

Luvo will be billed on the day that the change is made. The prorated charges must be based on the period 21 March to 14 April. I.e., 25 days x R3.33 = R83.25  
Luvo will be billed again on 15 April for R99.