There’s two parts to answering this:
1) Changing the tent: Go to the atomic shop. If you have any versions of the tent unlocked by your account, you can hover over them, and the game will give you a prompt to mark it as the active. Should work similarly to picking weapon or armor skins for Nuclear Winter.
2) Exclusivity of the tent: You must have Fallout 1st to have a survival tent. You must have had the F1st subscription active when the the APC was the monthly unlockable for subscribers. If you have first now, but did not have it then, there is nothing you can do to unlock the old tent variants. If you do not have 1st at all, you cannot even use the default tent.
