My experience with this game lead me to believe that any deaths during the Monster Challenges, whether to you play them in between levels or from the main screen menu, do not go towards your totals deaths during a chapter or for the whole game.
There are other posts on this forum that contradict what I've posted, but I know there were some challenges that I did over and over again to get gold. I did so in between levels, not from the main menu, and lost a good number of lives in those challenges. When I finished the game, I got the achievement for losing less than 10 lives during the course of story mode, something I shouldn't have gotten if the Monster Challenge deaths added into the Story Mode total.
This is not conclusive, so proceed at your own risk. Personally, I'd be pissed if I had to play through the story mode on this game twice. Not that the game is horrible, far from it. Just not a story mode I'd have fun doing a second time around.