OK, if you need to SOCKSify the traffic from 10.0.0.0/24 then the NAT is a wrong proposal. Instead you can redirect TCP/UDP traffic from the 10.0.0.0/24 to the local (running on the gateway) transparent TCP/UDP proxies which will handle the authentication with SOCKS5 server and forward traffic through it. I have mentioned the SOCKSIFY sample above which does this for the locally originated traffic, but the same can be easily done for the LAN traffic too.
Another component you may need here is DHCP server to assign IP addresses to the PS4 and XBOX, it can be a part of your software or external service.
You can use support(at)ntkernel.com if you prefer the direct communication.