Before I had kids, I would've said I preferred bags... but now I'm 100% jewelry/watches > bags. Jewelry and a watch are something I wear daily. When it comes to bags, these days I throw my license and a credit card in the diaper bag and I'm out the door.
I have a Chanel classic flap that I bought as my first big luxury purchase 5+ years ago, and it's honestly not gotten much use, especially since I had my first. I have the medium size, and I find it quite heavy. I adore this bag, but I don't see myself using it again for several years. I need something lightweight that can carry spare outfits, snacks, etc. and this ain't it lol
My husband got me a cheap, cute quartz watch before the birth of our second because I was ALWAYS asking him what time it was, and I think it's one of the best gifts I've ever received. When I'm nap trapped or feeding my newborn in the middle of the night (I take it off overnight but keep it right next to where I feed the baby), it's nice to know what time it is to loosely track naps, how long it's been since last feed, etc. In your case, I'd vote for the BB.
EDIT to add: congrats!! Parenthood is such a wild ride. Enjoy this precious time before the chaos begins