Re-wiring a router into the main control panel
MoCA might be a solution for you (Media over Cable Alliance). I note that you have quite a few coax cables in your LV box, does one of them go to near the spot you want?  If so, you can get MoCA adapters to repurpose the coax for use with ethernet.   You'll have to do some research, but this should point you in the right direction.  Here is a place to start:  https://www.signalbooster.com/blogs/news/moca
Disclaimer:  I have no personal experience with it, but it seems pretty well represented out there, your mileage may vary.   Maybe others here can comment on experience with MoCA.
EDIT:  This may be a question better asked at another part of SE:  More to do with computers and networking.   You are going to have other configuration issues such as DHCP controls using a 2nd router.  Why keep it? Do you have enough ports on the new router? If so, just use the MoCA adapters (if your coax runs to a desired place) and plug it in. If not enough ports, just add a simple (non-intelligent) switch to add ports. Multiple routers can be a head ache if you aren't an expert, switches will often do the job.
Thoughts from other about migrating this question?