Time Machine vs multiple routers issue

No. All 192.168.*.* addresses are considered non-routing addresses and will not cross a sub-net boundary.

When you have the router acting as a router, it will make all off-router access via NAT (Network address translation). And broadcast messages, such as the ones used by Bonjour for device discovery will NEVER pass through the router when it is acting as a router.

If there are features you prefer in the 2nd router, then swap with the first router, and put the original first router into "Bridge" mode, or whatever turns off it routing features. GPONModem>>Router2>>Router1

Or dump both routers and get a real Mesh router system with multiple mesh access points. There are several manufactures out there selling mesh routers. Find one that offers the services you prefer.

You need to put the 2nd router into "Bridge" mode, so that it just extends the existing local sub-net. This is sometimes done by a router option that says "Bridge Mode", and sometimes it is done by Disabling DHCP.

At the moment your 2nd router is creating a 2nd sub-net, and there are many local network protocols that do not cross sub-net boundaries, such as Bonjour discovery.