This is more than likely network related and unfortunately this often requires admin access to the network equipment to solve.
First thing you should try is prior to making the #1 call verify you are on wifi calling and turn off cellular from the control panel. Now your phone should not attempt to switch to cellular. I believe AT&T network is capable of handling the transition from wifi to cellular but there is a process involved for that transition with a lot that can go wrong that could be specific to your area.
If it still fails and just for testing you should disable wifi calling from other devices in the iPhone settings, reset network settings (yes that is very annoying so do that last resort), soft reset the iPhone and power cycle the router/modem (an internal conflict in a router can keep the internet on but not provide services for other functionality so its important its power cycled). Make sure your address is set in e911 in wifi calling settings and make sure location services is turned on for wifi calling in the privacy settings. Also just to make sure goto Settings > General > About and scroll down to "Carrier" and tap on that a couple times to make sure you have the latest carrier update.
The way Wifi calling works is your iPhone establishes a secure IPSec VPN Tunnel from your phone directly to your cellular providers network. This can be thought of as a direct cable from you to them for security and privacy (oh and so they can bill you when roaming).
The problem is for this to happen seamlessly for the user a few functions and features need to turned on and setup correctly. While some features need to be turned on like WMM others may need to be turned off like certain IPv6 and QoS features. The basics, are port 500/4500 udp and 143 tcp must be open. MTU 1500. IPSec access/authentication set to on.
Are you able to obtain that level of access to your offices network? Does your office restrict access to certain domains (ex. blocking vvm.mobile.att.net can lead to issues)? Does the office have multiple access points? And do you know your offices internet provider name and what equipment they (or your business) are using? When is the last time you think the modem/router saw a firmware update?
Often I will see an ISP supplies an all-in-one modem/router. Since this is often in a non central computer room connected to a network switch they will get another router to mount more centrally for better reception and run an ethernet back to the all-in-one. They check it and the internet works. However what they didn't do is set the all-in-one modem router to bridge mode so now they have "double NAT". The problem with double NAT is it will directly and negatively effect VPN tunneling, SSL, opening specific ports, etc its basically wifi calling kryptonite.
Anyway I hope something here points you in the right direction, good luck.
