The most common and most likely answer to this crossword clue is the 6 letter word NIPPLE
I've seen this crossword clue in Independent
N
I
P
P
L
E
More clues leading to the result NIPPLE
Buddy explains!

The straight part
We know that "bit of a tit" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case bit of a tit = nipple.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does NIPPLE mean?
NIPPLE noun- The projection of a mammary gland from which, on female mammals, milk is secreted.
- A mechanical device through which liquids or gases can be passed in a regulated manner.
- An artificial nipple (definition 1) used for bottle-feeding infants.
- Any small physical protrusion, such as the lumps on the F and J keys on computer keyboards.
- Any small physical protrusion on an automotive, a machine part or any other part that fits into a groove on another part.
- A perforated segment that fits into part of the breech of a muzzle-loading gun, on which the percussion cap is fixed.
- A short tube threaded at both ends, used as a connector.
- An internally threaded piece which holds a bicycle spoke in place on the rim.
- To fit (a baby's bottle, etc.) with a nipple.
- To give one's nipple to (a baby) to allow breastfeeding.