There isn't a way to change the payment details on your invoice.
If you have a monthly subscription, you can change the payment details by signing up for the monthly Pro subscription again right before the next payment is due. You can check when the next payment is due on your Subscriptions page.
If you purchasing for the first time, and are within the 30 day money back timeframe (after your first purchase on SoundCloud), then you can end your subscription here and get an immediate refund. You can then repurchase and re-enter the desired invoice details.