@Shubopady 

1. No. ER605 is like a regular router. It accepts a private or public IP.

From the subnet, 255.255.255.255, I can tell that this IP 10.247.116.156 is single usable IP in the ATT router. It means 10.247.116.156/32. Do you have another device using this IP? Subnet from the other router usually comes 255.255.255.0 10.247.116.X/24. meaning there are 253 IP addresses available. 

2. Try the unicast DHCP option? In the WAN settings, enable >> Get IP using Unicast DHCP

3. Change the subnet on your router to 255.255.255.0? 

4. Reboot the modem. I remember some modem will record the MAC of the first connected device. Make sure the router's the first plugged device to the modem. 

Those are my perception of your issue.

@John1234 

Thanks for the response.

1. No, I don't have another device using 10.247.116.156.  The AT&T router is in "IP Passthrough" mode and seems to always assign an address to my router like 10.X.X.X.  Since the WAN connection of my router is the AT&T router's only connection, it seems that 255.255.255.255 is OK for the subnet.

2. Tried unicast.  Didn't help.

3. Subnet on WAN side of my router should be set by DHCP, right?

4. Rebooted until there were holes in my boots.

Really, thanks for the response and the suggestions.  At this point, I think that there may be some unfathomable (for me, anyway) incompatibility between the old AT&T modem and the ER605.  I may just have to stick with the old router running OpenWrt.

Thanks again for trying to help, John.  Kudos.

@Shubopady 

:) You're welcome

Huh, strange. So, so it is passing through the IP from the ISP. What about contacting your ISP and checking if they have anything on their end? Did they record the MAC address and stop another router from connecting to their server? 

I think your ISP is using some kind of switch on their end and giving you a static IP(they configured on their switch). Since I deal with TP switch a lot. I know some kind of switch will learn the MAC address. 

@John1234 

Good suggestion to contact the ISP, but I don't have the 7 or 8 hours that it would take to get through AT&T's customer/technical support to someone that might understand the problem, only to be told that AT&T didn't intend for me to use a wifi hotspot for my primary internet service.  I've been down that road before :)

If I connect my laptop instead of my router to the AT&T box, my laptop gets a good DHCP assignment.  I can switch back and forth between my laptop and my old router.  I do have to wait a while (for caches to clear?), but both of them work.  Doesn't that eliminate the MAC address as the source of the problem?

Again, thanks for your responses and suggestions, but at this point, I'm giving up on the ER605 unless TP-Link tech support provides a solution.  If I do get a solution from them, I'll come back and post it here.

@Shubopady 

Like you said, you gotta wait for a while to get it working. I am still thinking it is recording the MAC address of the connected devices. What if you do MAC clone? Clone the MAC address of your PC or the router.

Which one is working now? Clone its MAC address to the ER605 and connect ER605 to the modem and test. 

User Guide instructions:

Choose the menu Network > MAC > MAC to load the following page.

So, if clone works, I believe that is your modem doing this headache to you. ISP will record the MAC address of the device and only allows the "remembered" MAC addresses to get Internet. To flush that, try to reboot, remove the power and battery of your modem, and call ISP to flush MAC on their end.