Yes, what you are trying to do is possible. It is however an unusual configuration which I expect to be unsupported by many routers. And it is going to be difficult for you to get confirmation before buying a router on whether it will support your needs.
A standard Linux system could do the job of the router. But I don't know if that's the ideal solution for you.
You are saying that the purpose of this setup is to avoid the need to have more than one Ethernet cable between the two locations. There is another and likely more robust way to achieve that which is to use a VLAN capable switch at each end of that cable.
From your question it sounds like you already have a VLAN capable switch at one end, and your current needs at the other end is a VLAN capable switch with at least 3 ports. Every managed switch I have worked with has had the necessary VLAN capabilities, and I know of managed switches with as little as 5 ports.
Between the two switches you will have the run of Ethernet connectivity which you wish to avoid duplicating. On that connection between the two switches you will need two VLANs. In principle you could run one of the VLANs as tagged and the other as untagged, but I think there will be less risk of confusion in the future if you tag both VLANs.
The new switch collocated with the router will then split the two tagged VLANs into two untagged VLANs on different ports connected to the WAN and LAN sides of the router. Any unused ports on that switch I would configure as untagged LAN ports, as that's what you are most likely going to want to use them for in the future.
Here is a modified version of your network diagram illustrating how the new managed switch would be inserted into your network:
Switch (VLAN and QoS for ISP)
New managed switch Router(AP bridge mode)
| | |\ |
Gateway router | | | LAN
| \_/ | |
LAN LAN LAN
A router to be used in bridge mode as access point (without using it's router functionality) is something which you can achieve with most routers by turning off the DHCP server and configuring the LAN IP of it to not conflict with the rest of your LAN.
Even if that router does not support bridge mode such a setup will still work as long as you don't try to use the WAN port on it. You can use any of its LAN ports for the connection up to the switch.