By just going off the public information about a flight and what one knows about an aircraft, is it possible to construct a differ version of a flight recorder's data well enough to fool investigators?
In order to construct an alternative Flight Data Recorder narrative good enough to fool investigators you would have to acquire a flight data recorder to feed the data to (this is the easy part). You would then need to feed the appropriate parameters to the flight data recorder to fill its memory (modern recorders can hold over 10 hours of data - this could encompass several flights that would have to be believably reconstructed - speeds and altitudes matching radar track data for example).
A flight simulator could probably be rigged up to supply the data to the recorder, so the primary difficulty here would be matching your forgery to other data sources like RADAR tracks, radio calls, the CVR (unless you intend to falsify that as well), etc...
Once filled with your falsified/reconstructed data the recorder would need to be injected into the chain of custody, which is perhaps more difficult. replacing one which has been recovered would require fooling the investigating agency, which has already seen, photographed, and cataloged the device they recovered (and would likely notice any subtle differences), or by finding the real recorder before investigators and replacing it with your forgery.
While it is not impossible to do all of the above it is highly unlikely that anyone would succeed in such a scheme.