The missing button is part of this purchase route
The publisher separates the website transaction from mobile viewing. Its FAQ tells iOS and Android users to sign in using the details created on kino-on-demand.com and find their website rentals in Meine Filme. It explicitly excludes renting directly through the app. Searching every app menu for a checkout can therefore be the wrong task, even when the app and account work.
Use one film to establish where the handover stops
Open the official website in your browser, find the intended film, and inspect the rental terms before making your own decision. After an authorised rental, return to the app with the same account and check Meine Filme. If the website library contains the film but the app library does not, that is a different symptom from a missing purchase button. Note that distinction when requesting help.
Do not buy a second copy to test the first
A browser receipt and a completed website library entry are useful evidence of the transaction. A trailer, a credit balance or an unfinished checkout is not the same evidence. Compare the account identity privately rather than posting the receipt publicly. The publisher asks for the film, device and browser when diagnosing playback; add the website-to-app mismatch if that is the actual failure. We have not made a rental or tested account synchronisation.