Is there a way to have Coreware sync batches with the Merchant Processor, so that we could easily run a report to match with deposits that were made in our bank?

For coreGIVING Non-profit:

 
There is a report called "Donations Report". This report allows you to order the report by date or bank batch number.

For coreFORCE Retail:

The Order Income report at Orders > Reports > Orders Income will show the detailed payment breakdown for each item you have sold in your coreFORCE retail store.

The Order Payments report at Orders > Reprots > Order Payments will show all payments in your coreFORCE system, including refunds, partial payments, etc.