As a trumpeter, I can say a bucket is similar, but not what you want ideally. With mutes that block the bell, projection is not as good, so by using a flugelhorn not only will it be a more authentic sound, but it will project and fill the room much better. But this would only really apply it was playing the melodic line I suppose? Tuning will also be better on a flugelhorn, mutes generally put off a trumpets tuning.
Depending on the question of who should play the flugel, it depends on the difficulty/flashiness of the music. If it is just relatively simple harmonic developments or a background motif or something like that, the 2nd trumpet is fine. But if it is front of house solo, then it should definetly be taken by 1st trumpet.
You question has allready been answered, but I think the point about projected sound may help :)