Restricting this to whether a native speaker would say, for examples,
add a little more sugar/ add a little bit more sugar
He's feeling a little better/he's feeling a little bit better
The use of 'bit' does not alter the 'amount'.
We would add 'bit' to stress how informally we are talking.
A chef on TV might say, "Now add a little more of the sugar". To exude even more warmth and friendliness, ' just like I'm a friend helping you make this dish at home', he might smile and say, "Now, add a little bit more of the sugar."
Talking to a doctor on the phone about her son, a mother might say, "He seems to be feeling a little better this morning."
If the doctor was in the room, with the child in bed, the mother might say,"He seems to be feeling a little bit better doctor" because she is also including her child in the conversation, as if saying, "He seems to be feeling a little bit better today...aren't you John." (said more as a statement than a question.