There are a couple of things you can try or use.

One would be using our Mesh network routers, for example the Deco M5 (3 pack). You can place one on each corner and one in the middle. Where AP 1 is located is the only one that needs to be wired to the router. The other two units just need a wall outlet for power.

If the original router/model is closer to the center of where you need coverage, you can enable the Wi-Fi for that device, and switch the 2.4Ghz channel to 1 so it will not interfere with the other two signals.

A simple range extender like the TL-WA855RE can be used to extend which ever AP has the stronger signal around the middle area.

You could also potentially use a powerline kit which uses the electrical wiring to send the internet signal through. A model to look at would be the TL-WPA7510 KIT.

Getting coverage over those middle areas isn't the problem.  Like I said, if I turn off either AP#1 or AP#2, the middle area gets service as expected.  It's when both APs are on, that issues begin in the area where the two zones overlap.

My understanding of Access Points is that, if set up correctly, devices should be able to connect to the one IP address seamlessly, since both APs have the same IP, and that devices in the overlap zone should be able to switch between AP#1 and AP#2 without any loss in service or degradation.  Is this not the case?  If it is correct, then these two APs aren't performing as expected.  Either the product is flawed/inferior, or there's some configuration that I'm missing or have incorrect.

Both AP devices must have different IP addresses.

Here is how they should be configured as far as a staic IP, note there is no need for a gateway.

Assuming your router gives IP address similar to 192.168.0.x

AP 1 should be 192.168.0.200, subnet will be 255.255.255.0

AP 2 should be 192.168.0.201, subnet will be 255.255.255.0

The wireless network names can be the same, and have the same wireless password.

Now, with the roaming or the ability to connect to different AP units without losing connection is not the case. Mesh routers (our Deco units) are designed to work with the 802.11k/v/r protocol which gives you a seemless connection no matter where you walk. Not all devices can do that or are compatible with those protocols. Here is a official microsoft article about about each one (https://docs.microsoft.com/en-us/windows-hardware/drivers/network/fast-roaming-with-802-11k--802-11v--and-802-11r).

The two AP units that you have do not do that. What it comes down to in your case is how your client device handles when to switch to another signal. The AP units have no control over that. There may be instances where your client device stays connected to AP 2, when it is physically close to AP 1 or you notice a drop in your internet mometairly for the reason that the device is switching over to AP1 from AP2.

If you want your device to connect to the strongest signal, that is something that needs to be configured in the client device. One thing to note is that when switching connections there will be a drop as you are connecting to a different network regardless.

Thanks for the feedback, Tony.

I know there are going to be challenges in this setup while moving through the coverage areas as in with phones.  What about in the case of my wife's computer, that is stationary?  If both APs have the same SSID and IP, shouldn't her computer stay connected to one or the other?  Why would it be constantly dropping connection?  Or are my expectations still unrealistic for stationary devices?

Both AP units should have the same SSID and not the same IP.

In your case the computer may just be switching back an forth. The computer is what will switch the connection back and forth as the AP would not tell the computer to connect to the other AP.

What I would do to avoid headaches is to see which AP the computer is closest to, and make a slight change that SSID so once the computer connects to it, it will stay connected, and not get confused going back an forth by itself.