How to set up a caching DNS server NOT on the router?

Question

Domain Name System

Good day.
I have a ZyXEL Keenetic Giga router at home and an old laptop on Ubuntu 14.04 as a server / file cleaner / test site. Internet comes to ZyXEL from the provider, and the router receives settings (IP, DNS) via DHCP from the provider. But since lately I have often encountered brakes on the side of the provider's DNS servers, I wanted to set up my own caching DNS server, dnsmasq, for example, on a laptop.
So, what I need is for clients connecting to the router to receive IP addresses and the address of my DNS server from the router via DHCP, but at the same time, for the DNS server to somehow magically receive from the router the addresses of the provider's DNS servers, which he provides. Bringing this magic to life is the problem. Can this be done, and if so, how?
I understand that you can not philosophize and just tell your DNS to use 8.8.8.8 and 8.8.4.4, but I want to .
Thank you in advance.
UPDATE:
Maybe I didn't describe very clearly what the problem is. How can I "forward" the DNS server data received from the provider to the laptop from the router, taking into account the fact that for all clients the router gives the DNS server address equal to the laptop address?

Answer 1

In the DHCP settings (most likely somewhere in the advanced ones), specify DNS1: Ubuntu gateway IP. Setting up a caching DNS according to any manual on ubuntu.

Answer 2

Well, if you really really want to RECEIVE from the router a list of DNS servers that the provider issued to the router, then this is only through a script - connect from Linux to the router with the console and somehow look in the settings what the provider gave him.
But somehow, according to observations, providers usually do not have a huge number of DNS servers, so you can simply enter them by hand, and that's it.