I think you need to make an extension where you modify the signature field from the biller to add tokens for the variable data you want to insert and then insert it yourself.
%custname% | %date% invoice from %billername% is attached.
You then parse this and replace the tokens with the needed values. You could do this in the modules/invoices/email.php file for the
Or you can do what I do. Use the signature field to say “Attached is your invoice for the current month.” The subject line already contains the invoice number and the biller name. And all the pertinent information is in the attached PDF.