Configure RB750Gr3 WAN ISP Switch

I have an RB750Gr3 (RB) which is working as a router for our LAN behind a VDSL modem in bridge mode.
My ISP gave me a fiber connection with a Cisco ASR920 switch, with internet access enabled on one switch port.
ISP told me that in order for the connection to work, I need a router connected to the specified port on cisco, with a (WAN) IP address and a gateway, and he gave me a subnet different from the one of the IP and gateway. The cisco is accepting traffic via NAT only via the specified subnet

We explained to ISP how is my config and he told me that I need a separate router to do the routing from my RB to the switch, and the subnet is required because in the future it is necessary to give also access to a different LAN router.
So the extra router will be my RB’s Gateway, and the address pool of the extra router will be the different subnet mentioned above with NAT enabled on that subnet.
The traffic according to ISP as I understood will be:

RB750GR3                                                EXTRA ROUTER                                 CISCO SWITCH
GW IP FROM SUBNET LAN      =>           SUBNET LAN / WAN IP   & GW               =>        INTERNET

How can we make the RB work with cisco keeping my local network settings and route via the subnet, without using an extra router? and keep my VDSL as a failover connection.

I’m not sure I understand what you wrote precisely, but there is no reason why you should need to put another router between your Mikrotik and the new ASR. You can use one Ethernet port of your Mikrotik to connect to the VDSL modem as a PPPoE client as it was until now, and another Ethernet port of your Mikrotik as a direct IP interface with a static configuration or with a DHCP client, whichever the ISP expects. If they gave you some public IP or subnet but they route it to you via some interconnection subnet, it’s still not a reason to add another router between the Cisco and the Mikrotik.

So if they e.g. told you that your public IP address is a.b.c.d and the interconnection subnet between the additional router’s WAN and the ASR is 192.168.1.0/24 with 192.168.1.1 at ASR side and 192.168.1.2 at the additional router’s WAN, you attach 192.168.1.2/24 to the Ethernet interface connected to the ASR, set a default route via 192.168.1.1, and instead of doing src-nat to 192.168.1.2, you will do src-nat to a.b.c.d. And if you plan incoming connections from the internet to a.b.c.d, you can use dst-nat matching on dst-address=a.b.c.d.

Or the a.b.c.d address may be up on some device in your LAN.

So provide a more exact description of what you want and what they told you, with particular addresses (replace the first three bytes of any eventual public address with A.B.C to stay anonymous) to get a more exact suggestion. Just for the case, 100.64.0.0/10 is not a public address range, it’s a “carrier grade NAT” range, not routable from the internet as these addresses are reused just like those from “private” ranges (10.0.0.0/8, 172.16.0.0/12, 192.168.0.0/16).

The “more exact description of what you want” includes the setup of the old VDSL connection as a backup. The Mikrotik can continuously (well, periodically) check transparency of the fiber uplink and whenever it is down, route new connections via the VDSL. Or you can prefer one of the uplinks for some connections and prefer the other one for some other connections, use exclusively one or the other uplink for yet other categories of connections… the possibilities are wide.