“date_paid” field set days after order creation

Is there a possibility to set “date_paid” field right after the order enters the status “processing” – which is when it is created?

Now the field is only set when the order gets status “completed”, which is after it is shipped from the warehouse, but that creates problem in tracing sales in other software (Hyros in our case), because the sale is tracked not when it happens, but days after it (when it is shipped).

For example, customer purchased on Friday with COD, order is created, has status “processing” and should be tracked by Hyros in that day and time. But it is shipped on Monday and only after that it changes its status to “completed” and the filed “date_paid” is set on Monday and this data is used to track the sale date in Hyros.