In some countries near Japan, such as Taiwan and Mongolia, there are sit-down flush toilets where you have to put toilet paper into a rubbish bin. Is there a term for such toilets? Would 台湾式トイレ (Taiwanese-style toilets) work?
Japan does not have the concept of a non-flush toilet. Even the port-o-potties at music festivals flush (including the TP).
Google searching "トイレットペーパー流せないトイレ" returns results such as "what countries have toilets that don't flush", "are there really places where you can't flush toilet paper", and "Why some foreigners don't flush toilet paper". None of them seem to have a word for it. Also, consider that high volume tourist areas have instructions (often in Chinese only) on how to use the toilet... not some single, self-describing term.
I think you should just call it like it is, with the phrase above.