You are likely going to have to do more testing to find what traffic is really causing the issue.
Verizon uses moca to connect the remote boxes to the router. It generally is not the bottleneck. If you get 350mbps speedtest results you do not have a issue with the moca. The moca is functioning very similar to a ethernet cable between your main router and remote box. If you plug you pc into the remote box via ethernet it is as close as you can get to ethernet to the main router.
The traffic never passes over the wifi network so it can not directly impact the wifi users. It would have to be either you are overloading the moca connection between the main and remote router or you are somehow over utilizing the remote router cpu or memory or something.
If you had said you were doing bit torrent file transfer then I could see it. But most online games use almost no bandwidth, like 1mbit up and down. So it can't be as simple as the traffic from the game is causing the problem.
I would watch the network resource monitor and see what traffic rates you machine is using maybe there is something else. Still you would have to be using all the 350mbps of bandwidth.
You might want to try local file transfers inside your house on the wifi devices to see if there is any pattern. Is it only internet traffic that is affected or do you slower file transfers between wifi devices during this time. Is it only device connected to the remote box or are wifi device connected to the main router also affected.
This is a very strange thing, the equipment you have should have no issues running wifi and online gaming.