Yes, if access point mode sets the correct settings then the PC will be in the same network as the quest and access point, as it should disable the local network on the TP-Link, leaving the Sky router to manage it.
Think of the Sky router as a town. It has two kinds of roads (wired and wireless), but they all interlink with each other so every person can meet with others in the town. Now think of the TP-Link, even if it has only one kind of road (wireless), if it's not in proper access point mode, as a completely separate town. Just like the Sky town, everything within the TP-Link town can see each other, but due to strict border controls on the main road connecting it to Sky, residents in one town can't see the other.
Setting the TP-Link to the correct access point mode will effectively merge the two towns together. If properly set up, the TP-Link's DHCP server should be disabled so anything on its wireless will be managed by the Sky router instead.
Different wireless names does not mean a separate network, you're just renaming the "roads" your devices connect to, so yes, keep it different so the Quest stays only connected to that.
Might be best to share screenshots of the TP-Link settings so we can help guide you to the correct settings.