The title says it all.
I would defer on this to anyone who has been a student at Yale in the past 40 years.
That said . . . I strongly suspect not. When I was a freshman, I was in DS, and the person with whom I shared a bunk bed knew that he was going for an MBB BS/MS (and later an MD/PhD). We both worked very hard, doing very different things. With one exception, I deliberately took fairly light, entertaining courses outside of DS (and even that one exception involved intense analysis and difficulty because everything was in French, but comparatively light reading and not long papers, because everything was in French). He did the same things with his non-science courses. He also spent a ton of time working in a research lab, which of course doesn’t show up on the curriculum requirements. He had a lot of capacity for work, but I think that three intensive reading/writing courses with nowhere to hide if you weren’t prepared would have broken him.
There was hardly anyone in DS who was thinking even of getting a BS in a hard science.
If you study the various requirements, I think it’s technically possible, especially if you can slim down the MBB prerequisites with AP credits, and limit your foreign language requirement to one semester. But you would be setting yourself up for at least three absolutely crushing years, probably four, with your first year being one of the worst of them. Don’t do that. (If you do the BS/MS, the third and fourth years are colossally crushing. In my day, very few people actually saw it through to the end.)
Remember that your uncredited research work is an important part of your science education, especially if you want a career in science. You probably also want to do things like take more than the minimum amount of math, and to make certain you have significant computer literacy. And you have to meet your distributional requirements.
It’s a mistake to get into the mindset that you have to prove you are better than everyone else by amassing impossible credentials, pulling academic stunts. You want to prepare yourself solidly for graduate study, really learn what you need to learn well, get integrated into the science research biz, pick up a mentor or two. Get yourself to the point where you can begin to express your creativity and intellect in research. You don’t do that by notching your gun barrel. Especially not with a credential no one outside of Yale College will ever recognize or care about.
DS is great, but it’s not necessary. If you want great humanities courses, there are great humanities courses you can take at Yale, many of them, without committing to have 2/3 of your freshman classes be DS. I loved the small seminars, and seeing many of the same kids in multiple classes (just like high school!), and most (not all) of the teachers were superb. But apart from having multiple classes with the same kids, you could get equivalent or better quality classes a la carte, and spread them over a few years.