Charging Interest on AR invoices

How does everyone deal with charging interest to overdue sales invoices in accounts receivable? I understand the process in sage, but am having a hard time with the amount of  manual work required to simply charge interest. It seems as though there is a large gap in accounts receivable. 

My first problem is that the interest charged on an account is only viewable on the customer statement report, and once you have that number, you have to manually amend an existing invoice, or create a new one all together. My second problem is that once a client pays an invoice - the customer statement report updates and the amount of interest for that invoice disappears (unless it is added to an existing invoice, or has been created in a new one). This would impact accounts where, say, you have a large outstanding receivables account with multiple invoices for a particular client, and that client can only pay in small lump sums over any period of time. In this example, you may apply payments to invoices, which may pay off those invoices entirely over time, but the interest for those invoices should still be outstanding! So if you have NOT created a new invoice for interest or added it to an existing invoice, then that interest disappears entirely, which should not be the case. My third problem is with the difficulty sage creates when trying to collect those interest payments! In our company, invoices are past due after 31 days. which means, every month, I send a customer statement report to all overdue accounts letting them know how much interest has accumulated and how much is now due. We also have payment options (like stripe and paypal) linked to our company profile, so clients can simply click "pay now" and be redirected to one of those sites to pay their invoice. The problem here, is that clients who click the "pay now" link through the emailed sage invoice are not allowed to enter in a different value to pay! So if they want to pay half of the invoice, they cannot use the link. If they want to pay the full amount plus interest - they cannot! They can only pay the exact amount on the invoice - and as previously discussed, unless we amend the original invoice to include the interest - its not on there! I like the customer statement report because clients can see the original amount and how much interest has accumulated. I do not like sending clients multiple invoices that have been updated to add interest! I would prefer for them to keep the original invoice as is and refer to the customer statement when remitting any amount - but the pay now links will have to be updated to allow clients to enter different amounts for this to work.  

Creating a whole new separate invoice for interest or manually amending an existing one to add interest is not reasonable. What happens if a company has hundreds of sales invoices a month that have all earned interest? The accounts receivable module in sage needs to be more automated - with options to accept or decline the changes (we may not want to charge all clients interest if there was a genuine mistake or error on their end). 

Are there any companies out there who experience that same problem? Do you employ a absurd amount of manual work for this work around?