I've been using the FIRE system for many years. No, there is no way to show a list of payers - that I know of.
What I do is print a report to pdf of payers with the software I created the 1099s with. If I have multiple groups of payers, I number each group. Then, upon submitting the group of payers to FIRE, you'll get a confirmation. The IRS calls this report File Statistics. I print that confirmation to PDF and use the same number within the file name that I used with the payers report I previously printed. This allows me to verify which payers have been filed and within what group.