One basic outfit you could make consists of a frock with shirt and cap. I have found a commercial pattern you can use in construction: Kannik's Korner KK-9001 "Infant's Clothing Pattern". Although this pattern is documented to the second half of the 18th century, as you can see from the Holbein portrait, with small modifications this outfit should be period appropriate for renaissance-era reenactors as well. I suspect that what appears to be a braid encircling the baby's head is most likely a cap made out of a fairly transparent fabric.
Detail from Hans Holbein the Younger. Artist's Wife and Children. 1528-29. tempera on paper on limewood.German
Full image found online here
Above two images: Details of Kannik's Korner Infant's Clothing Pattern KK-9001
Don't carry that "pastel baby nightmare" diaper bag... A large drawstring bag is simple to make. Or you could make a variation on an alms satchel for a shoulder bag.
Rialto thread on pouches
Rialto thread on period backpacks, bags and carrying baskets
A pilgrim's alms satchel: A pattern and method of construction by Stephen Francis Wyley
A sling for carrying the baby
Poppet or Baby - Renaissance "Barbie Doll"
Foemina Mediocris conditionis in Silesia (Commoner from Silesia) from HABITUS
LORVM, TAM VIRORUM QVAM
foeminarum Singulari arte depicti
Rialto threads on dolls
Garb the World - "Garb the World is interested in helping parents who want to participate in reenactments and role playing games with their children, but can not afford to purchase elaborate costumes that will be quickly grown out of." Mostly early period (tunics, trews, vests) garb at very affordable prices.
Maeve's Unmentionables - Custom hand-made clothing for Renaissance and Medieval re-enactment.
Fantasy Fabrications - We make children's garb as well as adult, and some of our products are available with applique. Any image from our library of traditional heraldic designs is available as applique, or you can provide your own image. We can also work with you to create a unique design for you.