We don't pick up the baby as a single movement. If we did, then yes there can be no magical weight change. But in reality, there's a bit of a choreography involved. Perhaps the baby holds up its arms, you put your arms below its shoulders, you raise it slowly. During this process, surely it can control how much of its weight you perceive?
Scientifically speaking, the weight of an object cannot change unless gravity changes or it's mass changes. The forces required to accelerate that object in a particular direction vary wildly based on all kinds of things that exert a force on the object.
Off topic, but that does make me curious how "weight" works in an environment where you're exposed to more than one gravitational field. Like if you're close enough to two black holes to be affected by the gravity from both.