Can I change my billing date within the same month?

Yes, you can change your billing date in the same month.

Illustrative example below: 

Luvo's current billing date is the 21st of March. On 20th March, Luvo updates his billing date to 23 March. 

Luvo will be billed for the prorated fee on 20 March for 21 to 22 March. Approx. 2 days x R3.33 = R6.66  

Luvo will be billed again on 23 March for R99.