When we use the transform API to transform our data into a new index, is there a way for us to set the document ID? (I think currently the default setting is randomly assigned some string as the document ID)
No, the document id can not be set as its important for the correct behavior when running in continuous mode. That's why the id is not random but deterministic.
TL/DR: In continuous mode transform needs to overwrite old documents with the new version. The same document id must be used, otherwise the destination would contain duplicates / different versions of the same pivoted entry.
If you run in batch mode and desperately need your own document id's, you could change them using an ingest pipeline. However, note that this is an unsupported feature.