I would consider more complicated solutions only if you first come to conclusion that these simple things are not fit for the purpose.
True that some fancy token based solution may reduce database load, but if the API is doing something useful then that one primary key lookup and potentially the password hashing won't be a show stopper. Drawback with tokens and skipping the DB check is that you can't simply kill a client behaving badly. With API key you can just change the key and requests stop immediately (with MVP product this might be an issue, since maybe you have decided to add rate limits etc later).