As the higher commanding officier aboard Voyager, Captain Kathryn Janeway perfectly knew the ins and outs of Starfleet's protocol & policy where the respect of the Prime Directive was its driving force so, it was her responsability to remind The Doctor that in keeping using the holo-ermitter after Sterling's debacle, they knowingly violated Starfleet's first rule of its policy & could be held accountable once the news would be known ...and informed him that she had to confiscate the 24th Century's device (I bet she didn't even inform Starfleet HQ when the contact was made in S4 because if she had done, she would have been forced to confiscate immediately, regardless the Quadrant where Voyager was!

), no matter The Doctor agreed or not.
In fact here, she played ease in pretending to ignore where the holo ermitter came from and agreeing to let The Doctor using it for the so-called good of the service (for the same reason, she could justify his sentiency) while in fact, it was for their "ease" and for The Doctor, his pleasure, which from my point of view, was much worse than all her other violations, which I remind you, were made in the moment with the sole purpose of saving Voyager, her crew and sometimes, other populations from a total
annihilation. Not that these last violations were less serious but at least, they were justifiable or alternatively, understandable* par admirals, who once were Captains.
(Still in my mind, the only violation committed by Janeway which was not justifiable and difficult to understand by people from outside (=Admirals) was the one commited in Endgame, where she agreed with her older doppelhänger, Admiral Janeway, to use again a device from the 24th century to go home, even if her decision was explained by her loyalty & emotional attachment to some members of her senior officiers.