There’s a number of different options ranging in complexity, setup time, and hardware cost in my mind. For starters we assume that the only requirement is that you need your Switch to be able to reach the game server running in your house. A lot of what you can/can’t do with your router may determine how much additional hardware is needed in this setup.
The first issue to address for consistency is your IP address, which is probably set dynamically by your ISP. This makes anything you do difficult. I would set up Dynamic DNS (DDNS) - you can do this for free with a service like DuckDNS, and many routers have an in-built option for updating the DNS server with your current IP if it changes. This basically lets that IP address change around while you hit a consistent domain name (for example, if you use DuckDNS, it could be creativemode1352.duckdns.org).
Once you have that consistent address to your house, you need some way to get in, which is where a VPN makes the most sense. By using a VPN tunnel, you shouldn’t need to forward any ports to that specific game server host… although I fail to recall if the VPN server needed a port forward to its listening port. The VPN server would be in your house.
It’s not completely easy, but I use Wireguard and it works well enough (again, free). If you don’t have an Ubuntu/Linux box lying around then you could run it in a VM. Disclaimer: you may need to set up static routes out so that when you full tunnel, you see those devices. Hopefully your router has a setting for that.
If at this point or anything before your router isn’t working, then you may want to purchase an actual router/firewall appliance, or build your own (I.e. with OpnSense, pfSense). You can set whatever the ISP gives you to be in Bridge mode so that it just functions as a simple modem. This may or may not require buying a separate Wireless AP as well.
From there you can remote into your house, but there’s a bit of an issue here… the Switch doesn’t have a VPN client on it. In this case, you broadly have two options: use an existing device as a mobile hotspot (Windows PC, phone), or get a dedicated device that essentially creates a mobile VPN wireless access point. Whatever that device is, the idea is that it VPNs into your house, and just allows your Switch to connect through it.
As icing on the cake if you get this all set up: you can manage the power state of your physical server through Wake on LAN, if that PC has a hardwire network card to your router. That will allow you to boot the PC from sleep/shutdown, and you could have startup tasks to initialize the game server.