That is the best route documented in the sources reviewed for this page, but it does not state a price and does not prove that the search phrase “second-story key” is the exact in-game name. No coin value is stated on this page: the sources reviewed for it do not provide one.
If the key is absent, confirm the current journal objective, neighbor/task progress, reputation level and shop category. If ordered but missing, check the parcel terminal before buying again. After unlocking the room, the walkthrough places the Genetic Lab project there, tying the key to later lab progression.
A ready version needs a 1.0 capture of the exact key name, shop category, price, prerequisite task/reputation, parcel delivery, door location and unlocked content.