Hi. Why would you want to record invoices in Odoo AND in your other accounting software? If you only record invoices in Odoo and don't book any banks then why would you want to have the invoices on a paid status? You don't use it to perform the payments because for this you use another system.
You can set the invoice on ''in payment'' but I don't think, other then creating a MISC journal, there is an option to put it on ''paid''.

Thank you for your reply. Actually, we use Odoo for automatic invoice entry and validation by different people.
Once the invoices have been validated, we export the entries to an accounting software that is connected to our bank accounts. So I'm looking for a way to mark invoices as "paid" in Odoo, once they've been paid.