Our first two babies were baptized in church the respective Sundays immediately following their respective births. It was awful. Getting the baby and my terribly sore self clean and church-ready, ensuring that the baby's belly was full at all the right times and that no one was offended during the process, standing through the lengthy rite, the crowd of well-wishers needing food and a place to go, and pretending that I wasn't about to succumb to hysterical postpartum weeping the whole time were not challenges I had any desire to revisit by the time #3 came around. Furthermore, we never liked the 45 minute drive home with our precious little unregenerate heathen and living nervously with him/her until Sunday.
So #3 was baptized in the hospital 7 hours after she was born. Her godparents were local and were able to attend, the family members we had in town came also, and I watched the whole thing from the relative comfort of my shapeless gown, unwashed hair, and the bed in which it had all gone down. Her baptismal gown was presented as part of the rite. She wore it a few weeks later when the sponsors were enrolled on a Sunday morning at our church and we took home the baptism certificate.
We did the same thing a few weeks ago. The sponsors weren't local this time, but one of them was able to join us over the phone to witness with his ears and say his part. The only family members available were our three other children, accompanied by the nice lady from church who took care of them that afternoon. I don't know when we'll enroll the sponsors this time, and the little dude may well have outgrown the beautiful gown Grandma made for him by the time we get around to it, but I wouldn't ever go back to the old way.
It's an unconventional method at this moment in history, but it is so much gentler on the people who really need it: the baby and the parents. It would probably have been a harder sell to our family on our first baby; lots of them wanted to be there for that baptism (although we wouldn't have put it off if all interested parties hadn't been able to make it). The congregation misses out on witnessing the baptism, which is too bad, but doing it this way demonstrates to them that the baptism itself is much more important than the sentimental contemporary social customs surrounding it. Anyway, the point is, there are options. Your baby's baptism doesn't have to be a horrible strain on you, which is the last thing you need in those early weeks. It's one of the best moments in a parent's life, so don't let it get ruined for you. (Maybe the Gauntlets would like to share the approach they took with their 3rd, too.)