Okay, ive read your post a few times and i think i got a grasp of the situation. First, your pic doesnt work. Can you direct-link it rather than linking to something in reddit?
Until then, have you tried using the proprietary Y cable between the Asus Router and the patch panel rather than the standard CAT5s you’ve been trying? There’s obviously some proprietary thing going on between the patch panel and that Y cable. You didnt say where all 3 ends of the Y cable go, are there specific blue and green ports for it on the patch panel? Does the patch panel have any branding or model numbers on it?
The RJ45s (normally) are clear so you should be able to see the pin layout with the wire pairs, that could also tell us what is going on.
Probably not the “best” as its a few years old now, but ive been rockin an Asus RT-AX92U setup for a while now and it had great reviews when i got it. No issues, tons of coverage, and great features. I’ still using the stock firmware, but i know it can be better with Merlin, i just havent had the need. Supports 1gb wired backhaul and can have as many nodes as you want. You can get a 2-pack for like $320ish i think? Then just buy a 3rd single and your well within your budget….assuming this router meets your requirements otherwise!