It just seems weird that they wouldn't want to keep it supervised. With the cloak in starfleet's posession they could easily just dissect the thing and figure out new weaknesses in it, and how to exploit those weaknesses to gain an advantage on the romulans. I end up rationalizing it in my head as "The romulans gave them a shitty 20 year old out of date model of their cloak, so they dont care" but i don't even really buy that.
Starfleet developed a cloak that can pass through walls/asteroids/whatever. Merely becoming invisible is soooo 23rd century
I think the advisor wasn't to keep an eye on the cloak itself, but on the application of it - basically to keep Starfleet honest that it was really only using it in the Gamma Quadrant - the Romulans sure wouldn't stick with such an agreement without strict supervision. Still doesn't explain why she disappeared without explanation, though.