@David-P 

As you are aware the heavy plaster walls are a problem for wireless signals.  Connecting two routers together generally isn't recommended.  If you do the 2nd router, the one not connected to the internet source, would need to be in AP mode.  This way all clients are technically in the same network.  However, in AP mode, it would still generate a separate WiFi signal. 

The best solution, if you want the same SSID through out the home, is to get a Mesh System like our Deco.  Mesh is essentially multiple routers working together to form a single network.  The units can be connected to each other via a wireless backhaul, but also in situations like yours through a physical Ethernet cable.  Using Ethernet as your backhaul usually improves transmit rates so its actually recommend when possible.  The benefit is that all units in your mesh network can transmit the same SSID.  And you can add to them easily when expansion is needed.  

That would be what I would recommend for you.  If you don't need WiFi 6 or 6E then the Deco S7 is a good choice.  If you do the Deco X55 or Deco XE75 are also recommend models.

  @David-P Putting the 2nd router in AP mode and using the same SSIDs and settings is fine for most.   I've been doing it for years without issue.

Getting a mesh system by itself won't make your wireless better.  What matters is the "fast roaming" feature - https://community.tp-link.com/en/home/forum/topic/180194 .  The Decos include fast roaming and the standard routers do not.

A decent video describing issues and suggesting separate SSIDs as a possible alternative:  https://www.youtube.com/watch?v=kf1mPd_ofhU