If someone had a gun to my head saying, don't hide the sub-penny slices from the billing, I would just do the billing in thousandths of a cent, rather than make "leap cents" appear every couple of bills:

  Amount owing: $123.45678

  Please pay one of: $123.46 (a credit  of $0.00322 will be applied)
                     $123.45 (a balance of $0.00678 will carry forward)

On the next statement, if they paid $123.46:

  Previous balance: (  $0.00322) [credit]

  New charges:        123.45678  { here we have a detailed breakdown }

  Amount owing:       123.45356

  Please pay one of: $123.46 (a credit  of $0.00644 will be applied)
                     $123.45 (a balance of $0.00356 will carry forward)

etc.

That's literally "put the billing detail from the 3rd party API and assemble it into a statement". Since the billing detail from the 3rd party API is in thousandths of a cent, then that implies the statement must have thousandths of a cent.

If the two payment options were determined to be too confusing, one of the two could be dropped.