Ok - figured it out. Since the client machine could not reach the urbackup server because the urbackup server’s public IP address changed I have to do some network funkiness.
I added the old public IP address of the urbackup server as a secondary IP on the client server’s NIC. I then used netsh portproxy to listen locally on the client on ports 55414/55415 and forward any traffic to the new public IP address of the urbackup server. This allowed the client to connect. I then changed the urbackup server configuration so that it used a hostname rather than an IP address for the Internet server. Because the client was connected, it updated the config on the client to use this name. I then just created a DNS entry for that name pointing to the new IP address and the client is back online.
I then remove the netsh portproxy entries and the public IP address from the client NIC and everything is back to normal.