My DDs are turning 4 later this month so I finally brought out my HAC 32 (similar size to B35) to use after 4+ years hiatus just this week. It is heavy when filled and that's why I haven't used mine from the time I was expecting. While yes it can be hung on a pram but because it doesn't have a shoulder strap there will be times when you'll need to carry it by hand or hang it on your forearm or put it on the floor while holding a baby so there's a slight inconvenience and a potential for getting the bag dirty or scratched. I ended up using an LV neverfull GM but most of the time I used the Longchamp Le Pliage because it weighs nothing.
The bigger the birkin the heavier it is. As lovely64 suggested, may be see if you can try it out first. When you have a baby and a stroller, things can get cumbersome.
As for usage, you may want to consider ordering a bag organizer (there are specific ones that are diaper organizers--check etsy.com) to keep your stuff from moving around (I use a regular organizer to keep my leather interior from getting scratched). For day to day you'll probably carry diapers, change pad, wipes, bottle(s)/sippy cup, snacks (when baby's older), extra change of clothes, burb cloth/mini blanket, a soft toy/rattle, hand sanitizer, your own personal items like wallet, sunnies, phone, water bottle, etc. G'luck and congrats!
