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The description says the readers are POE. They are most likely hardwired to a switch unless there are inline POE injectors. Checked if the switch supports mirror ports. You can mirror either one POE reader ports or uplink port. If the switch doesn't support mirror ports, try to insert either a hub, switch with mirror ports or TAP inline with the uplink port. It is possible to do the same on the POE reader ports, but you must ensure the readers have power.
After you have a capture, you can filter for ping requests and replies. The replies packets will have the response time in milliseconds. If you want to be creative with the display filter, there are several you tube videos demonstrating how to analyze pings with Wireshark. The syntax for Wireshark display filter can found at the Wireshark Wiki DisplayFilters .