A Linksys router should work fine. Pretty much any router should work fine.
A router isn't the only way to get it online. You can get the USB LAN adapter, which all said might be cheaper and certainly less work to get it online. Plus you eliminate any chance of interference.
That said, if you still go with a router it has no IP address that makes it look like a router. The 192.168.x.x range is what it presents to devices that connect to it from the internal side (the side your Wii will be on). The external WAN side can still accept an IP address from your school's network via DHCP (or it can be set manually if your school insists), but either way the IP address range is set by your school so it's up to them.
The MAC address can only be traced to a manufacturer. So while someone determined might look it up and see that it's "Linksys", they can't tell if it's a Linksys router or Linksys network card. But I highly doubt they'll bother.