I was wondering if it’s somehow possible to configure the Flint 2 in a router mode (with all the features it allows) connected to existing router which is connected to the Internet, and I want be able to connect LAN computers from backbone to the Flint2 router?
Technically you could do it but it will be a PITA for long term management. If you plug the primary/upstream router's LAN cable into the WAN port of the Flint v2 you'll be double-NAT'd. You would have to set up port forwarding for every service/daemon downstream within the 'inner' subnet (192.168.0.0/24 — routed by the Flint v2 @ 192.168.8.1) if you want the primary LAN to have two-way communication within the downstream/inner subnet.
See the following thread for an example of how much of a mess it can create: