But I would also like to connect wirelessly to my Archer C7 with a different SSID than the one they use upstairs so I get the best local connection between my own devices. Is there another guide I should follow for this? Or what do I need to do to achieve this?
There are two radio interfaces active. The "incoming" one is a client and must be set to match the network upstairs. The "outgoing" one is an AP (master) and you can set its SSID to whatever you want.
If you only need the network upstairs for Internet access I'd suggest a routed client instead of a relay. The routed client will firewall your network from anyone on the LAN upstairs.