Can I change my billing date shortly after I was charged?

Yes, you can.

Illustrative example below: 

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

Luvo updates his billing date on 22nd March. 

Luvo's new billing date is now the 25th 

Luvo will be billed the prorated charges on 22 March for the period 21 – 24 April. I.e., 4 x R3.33 = R13.32  

Luvo will be billed again on 25 April for R99.