We know that Hobbits were 3 to 3.5 feet tall, Bandobras Took almost 4. But how big were hobbit children (relative to their age, of course)?
For they are a little people, smaller than Dwarves: less stout and stocky, that is, even when they are not actually much shorter. Their height is variable, ranging between two and four feet of our measure.
-The Fellowship of the Ring
Since Tolkien says here that Hobbits aren't particularly out of whack in terms of their proportions, unlike Dwarves, I think we can assume that they have human or near-human proportions, just shrunk down to about 3/5 of our size.
If they have human proportions, then you can assume their children would also be 3/5 the size of human children at their respective stage in development (keeping in mind the average lifespan of a Hobbit is longer than a human's.)
Hobbits: 3' - 3'6" Humans: 6' - 7' Human babies: 1'7" - 1'9"..
Since hobbits are virtually identical in anatomy to humans (only smaller), I think it's safe to assume, hobbit babies would be roughly 10" (edit my typos, if you must, but don't change the values I provide!!!), a little more than half the value. Is there any canon reference to this?