1. EMH Mark II's answer was annoying to begin with. A simple "no" would have worked.
2. Even if we assume that Voyager's computer system was updated with information about the Dominion, and then assume that EMH Mark I had access to the wealth of knowledge in Voyager's database (which doesn't seem to be the case anyway) .. is it really reasonable to believe that B'Elanna would have shipped the entire library of Federation threats along with the Doctor's program over the Hirogen relay given the risk of sending just the Doctor? He also obviously couldn't pick up that information from Prometheus's database.
I personally think we should take the Doctor's word for it that he never heard of the Dominion and accept that sometimes characters know less than the viewer because the viewer gets to experience everything, whereas the Doctor wasn't even turned on until the DQ.
Edit to add, slightly off topic:
Janeway did have another option. She and Tuvok could have sent Voyager home (The way Tuvok was talking, he could do it at a moment's notice), with the two of them remaining on board the array until the last possible second to ensure the explosives went off.