You could always get a seat for one in front of you (handlebar mounted child seat)
and the second behind you (conventional rack-mount child seat). Probably have to have them trade off to minimize fights, since the front seat is probably a lot more fun (and easier to watch the kid). I see a couple around town where the wife carries their kid in a handlebar mounted seat that looks just like that one.
Or you could get an Xtracycle and get the "peapod" accessory to put two kids on the back of your now-longtailed bike. There's probably similar options for other longtailed bikes, or simply seats that you can mount two of two a longtailed bike... Here's a Kona Ute with two peapods on the back:
You should check the age, though... I understand that you want to wait until a couple of months after the child can hold their own head up (the helmet makes it even harder on the child's neck, but small children really need to be wearing a helmet). Something like 9 months? Or maybe find a seat arrangement that has the kid laying down instead of sitting upright.
For a solution that's compatible with a tag-along bike, you probably need something that puts both kids in front of you. A bakfiet's probably way too expensive, though.
I see lots of parents carrying kids on bikes, but only ever a single kid on the bike, if there's 2 or more kids, all the others are in a trailer, on a trail-a-bike or on the other parent's bike.
Disclaimer: I don't have kids and have never tried to carry even one on a bike. Closest I've ever come is either a lot of groceries on the bike or an adult on a trailer.