Have faith in yourself and the kiddos. 4.5 months is more than enough and many folks have done it within much less time. When we decided to go with BS, we had the campus tours/interviews in October, did additional research in November, wrote the essays and submitted our packages in December. It all worked out just fine.
Most schools use common application systems like SAO and Gateway, so you will only type your info twice and not for every school you apply to. Very few schools require essays specific to them (Groton, SPS and Deerfield come to mind), so the amount of work really isn’t that much. Better yet, this will be a great practice for college applications!
It is interesting that you mentioned the maturity of kids featured in the promo videos/websites. IMHO, it was only natural for the admin office to pick the most presentable ones.
Full disclosure - I did not think I was ready when my dad pushed me to attend BS years ago. I was doing very well academically at a prestigious private school near our home, but all we did was study. My dad thought I should leave for BS to be challenged on all fronts, especially from a social and diversity perspective. Boy, was he right. While I struggled during the first 2-3 months in BS, I gradually adjusted myself and blended in just fine. I made amazing friends that I still hang out with today. I picked up new sports. I attended small, thought-provoking classes. I learned to present my own views effectively and engage in friendly debate. Those would not have been possible at my local school because the focus and student/faculty ratio were quite different.
Having said that, BS are not for everyone. You really need to visit them to get a feel. Finally, as I mentioned earlier, leave the ultimate decision to your kids. They are the ones going to school, not us.
I hope this helps. Good luck either way!