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I would develop it using 2.2.3.
If you want to ensure that it's compatible with 2.0.13.1, then test it with that version too, either during or after the development.
Within the module, don't access any database tables directly (except for your own module's tables). Use the XOOPS core classes instead. For example, if you need to map a user ID to a username, use the member handler class, rather than doing a query on the users table. That will help ensure compatibility.