Governments in general are entitled to do things they forbid their subjects from doing. Gathering taxes, jailing or shooting people who don't behave, performing genetic experiments...
Police forces today dabble a lot in small scale drug manufacturing, for purposes of developing their methods of detecting the forbidden substances. Doesn't mean that they would be pro-recreational-but-lethal-drug-use, or acting behind the backs of their superiors.
Darwin is officially named a Genetic Research Station, so secrecy apparently isn't an issue. Picard is filled with "apprehension" when considering possible ties between the Lantree
deaths and genetic research, but similarly Kirk is filled with apprehension when thinking about penal colonies in "Dagger of the Mind". Both may be thinking of bad old days there, even if only Kirk is corrected on this by McCoy; Picard may well be capable of doing his own correcting, even if he enters his reactionary thoughts into his log as an act of personal defiance.
Also, Picard is apprehensive about a connection with "a" genetic research facility, suggesting the Federation has plenty of those...
I actually thought it made some sense, but it becomes problematic when you can always heal people by using untainted DNA.
Also, they blew it by having Pulaski come out of the transporter treatment fully up to speed. Had there been more cuts in the scene, we could at least pretend that the treatment "reset" Pulaski so that she would remember nothing between now and the day she last brushed her hair. As matters stand, the only cut where Picard could brief her on what just happened would require the two to stop in the transporter room doorway for a couple of unfilmed minutes...