What Web Access is measuring is the round-trip for an LFSO call to the Laserfiche server.  So the operations are quite different from a network perspective -- establishing a TCP connection and sending multiple data packets vs sending a single ICMP packet.  Ping essentially gives you a minimum value for round-tripping data to a server, and that KB you linked describes a number of additional reasons why WA might measure something larger.

The next trouble-shooting step would be to use a tool like Wireshark to verify the timing reported by the configuration page.  You should see the outgoing request, a delay, and the incoming response.  The delay there should be pretty close to the one WA is reporting.  You could at the same time run a capture on the LFS machine.  If the delay as measured from LFS is smaller the problem is in the network.  If it's about the same, you'll want to investigate why LFS is so slow to respond.