Since this looks like a client side problem, I would look at the route table of the PC after successful connection into ASA. If you are using split tunnels, the ASA and Anyconnect client installs routes in your routing table for all the split networks. What I would look for is a conflicting route for that server ip address.
Another possible avenue for troubleshooting is to apply Wireshark captures on the Anyconnect and physical adapter when testing pings to the servers. You should see traffic go through to the VPN adapter is the rules are correct.
