Just get rid of the whole "top secret" mission thing. For Starfleet to want to check on the status of the Borg makes sense, if they can figure how to get a ship there and back in a reasonable period of time. Less than a century round trip would be nice. Figure a way for that.
Send a regular Starfleet vessel, something mid sized. A Defiant is too small for a mission lasting several months, maybe a couple of years. there are a lot of star systems to check.
Plus a Defiant lacked the survey equipment for such a mission.
Starfleet likely wouldn't want to tie down a Galaxy, too much of a asset, A older Ambassador type (or smaller still) would do nicely. Again if you can figure how to get it there.