The defining characteristic of US classified documents is that their release would cause some degree of damage to US national security, ranging from harmful to gravely harmful. here’s a Cornell Law writeup that squares with what I know here.
Regardless of any opinions one might have as to the use and application of classification, in the eyes of the US government taking these documents without authorization is harmful by definition.
My mental justification is that the Tardigrades were either extinct at that point (which is stated to have happened at some point prior to the end of DISCO S3), or what few post-Message in a Bottle admirals might know about the classified project couldn’t justify making them suffer-- because, y’know, that is a thing that happens with the spore drive.
Hell, even if it was an option, would Janeway go for it? We saw her get rightly pissed at the equinox crew for running their ship off space aliens’ suffering. and I feel like the next-closest alternative known at the time (genemodding someone with Tardigrade DNA and also making them suffer through the jump) might also fly in the face of her highly principled stance.