Is it possible to print AP checks by batch? Or is there a different solution?

Our client enters their AP Invoices by batch.  They also want to Select and Print their checks by batch.  When I go to Invoice Payment Selection, by batch is not one of the choices.  Is this possible?  I cannot seem to find this option and am wondering what I am missing?  They are also asking if they select by Invoice Due Date, can they name their check run by batch?  So, if several people have a check run on the same day, can they give their own check run a Batch ID, so they can search for checks in the future by Batch, or print and post the Check and Electronic Payment Register by Batch.  So, I guess the main subject of my question, is what Batch features do I have available when selecting AP Invoices to pay, and when I print checks for those invoices?