There is a website that offers both a codebase to process some arbitrary public data (as Apache 2.0), but licensed the output data of the workflow as BY-NC-SA 4.0 (at least the processed data on the website). Can I re-produce the output data with the (free) codebase, and integrate it into a commercial offering? Presumably, the data is considered a derivate of the code, and should be covered by Apache 2.0
IANAL/IANYL, but your presumption is wrong; the output of a program is not generally subject to the licence of the program. Instead, the output data are generally considered a derivative of the input data.
So in this case, as long as the input data aren't subject to any kind of restrictive licence, you should indeed be able to download a copy of the Apache-2.0-licensed code, run it yourself, and recreate a copy of the output data which doesn't have a restrictive licence attached. You don't give us any information about the input data, so it's hard to say anything more precise than this.