Did you contact your ISP about this? Most of them can adjust a setting for you to remove the NAT part, the feature is usually called dual-stack. If you are in the EU, you even have a fundamental right to use your own router, you just have to register your MAC with them.
I am not extremely familiar with nix and flakes but from my understanding, they create isolated environments like venv or containers, correct? Would that not mean having a separate wine install and especially separate Lutris install per Game? So no single Lutris that lists all games but a separate one for each, kindof defeats the purpose I think. I fail to see the benefit over just clicking it on the lutris website, so not sure if this would gain traction. Open for any suggestions and corrections though!