I did a reading three weeks after my baby was born. Sure, I was still bleeding, but no-one suggests you can't attend a wedding because you've got your period!
OP, how many other bridesmaids do you have? If you have "several", I don't see why you can't say to the others that your pg bridesmaid is excused from other duties, and will wear the frock and walk down the aisle etc without having to do much else? If you feel there's more work than the remaining bridesmaids can do (although mine just supplied champers and kept me company) then recruit another. If you've got a big do, we're only talking an extra £100 for the extra dress. If your others can do these "duties", let the pregnant friend have a free ride.
Dress wise, I suggest she orders it in the next size up. I was one of those irritating types who lost it all quickly but was still a size larger three weeks after. And get wraps - at least for her, if not for the others. If necessary, she can safety pin the back together and strategically position the wrap around her back
For the baby - s/he'll probably be in a car seat - is there really no-one who can sit with the baby, just in the ceremony, and be in charge of removing them if the baby kicks off? Three week old babies really don't care who they're with as long as they're nice to them. An aunt, a trusted friend?
And - don't suggest a reading as an alternative. You can, actually, do without one bridesmaid, but although it was an honour to do my lovely friend's reading, I was concerned that if I had a difficult birth and was unable to make it, that I'd ruin her day by having a stand-in (wrong name on order of service, no time to rehearse, etc)