Is it appropriate to "shop" through high-impact journals before sending the paper to more specialized journals?
Judging the value of work is subjective, and can differ significantly between different researchers.

There is not harm in trying high-impact journals if some experienced people (like your mentor) consider it worth-while. But if you tried three different journals and get more or less immediate rejections, you should probably aim lower. Otherwise you just waste time in waiting for another five rejections (and you also waste the time of the editors/reviewers). 

Maybe, after three rejections, you can discuss this topic with your mentor again, mentioning that you don't want to wait another three years for your paper to be published.