Question about AVM FRITZ!Box 4040 International

The instructions state "of your other / existing router", i.e.: main router of the third-party provider, in your case "Salt Fiber Box".

Setting up FRITZ!Box as an IP client
https://avm.de/service...
When operating as an IP client, the FRITZ!Box uses the existing Internet connection of your other router. In this case
the firewall of the FRITZ!Box is deactivated.
the computers, smartphones and other devices connected to the FRITZ!Box continue to receive their IP settings from the existing router.
All devices can access each other without restriction, regardless of whether they are connected to the FRITZ!Box or to the other router.


Aside:
According to the descriptions in the various AMV Fritz! products, MESH is only possible with Fritz! models that have FRITZ!OS 7 (and higher) installed.

Example:
The FRITZ!Repeater 3000 has two types:
Variant Set up LAN bridge
Activate MESH variant (only with FRITZ!Box from FRITZ!OS 7)

Page 21:
Requirements for operation
For operation with LAN cable: Router with free LAN port [router is not specified in more detail].
For Mesh: FRITZ!Box with FRITZ!OS 7.00 or newer
https://assets.avm.de/files...

P.S.:
At AVM you have to search in the downloads and in the knowledge base.
If something doesn't work, then Salt customer service.

Now it is clearer: in addition to WLAN in the living room, LAN for one or more devices in the living room, LAN cable between office and living room already available:

this can also be easily solved with AVM FRITZ!Box 4040 (via MESH)
The "cascade" variant is important for you.

Connections
WAN connection for connection to cable/DSL/fibre modem or existing network
See picture at:
https://avm.de/produkte...

How to set up:
See chapter: Setting up FRITZ!Box for operation with another router.
The FRITZ!Box can share the existing Internet connection of another router. This allows you to use the FRITZ!Box
use the FRITZ!Box on an Internet connection where the Internet service provider requires the use of a special router supplied by him with permanently registered access data (Integrated Access Device, IAD).
Integrate the FRITZ!Box very easily into an already existing network infrastructure.
You can set up the FRITZ!Box as an IP client or as a cascaded router.

Select the "Mesh Repeater" option.
https://avm.de/service...

Perhaps combine with FRITZ!WLAN Repeater 1750E ("Mesh" method)
Prerequisite:
Office and living room must have a direct connection:
either via WiFi (based on the description: not possible)
or via LAN cable then FRITZ!WLAN Repeater 1750E
or copper cable (power cable) then Powerline from e.g. Fritz!

AVM does not suggest linking 2 routers (see pictures on MESH: WLAN from S to XL
Only one router is possible per address, otherwise a second subscription from the provider is required.


https://ch.avm.de/service...

4 Connecting the FRITZ!Repeater to the FRITZ!Box
Connect the FRITZ!Repeater to a LAN port of the FRITZ!Box using a network cable. If a direct cable connection is not possible, you can bridge the distance to the FRITZ!Box with a Powerline connection (e.g. with FRITZ!Powerline).

Further details
http://ch.avm.de/produkte...

http://ch.avm.de/produkte...

Enlarge your WLAN. With Mesh!
https://ch.avm.de/mesh

Perhaps you can solve the problem elegantly with products from the "Powerline department" and save yourself several metres of LAN cable.
See, for example, the Devolo product descriptions (manufacturer's page). Devolo WiFi (!)