In a normal world and with sane people it may work but we are dealing with Westminster MPs here.
They have proven that if they have more than 2 choices, they cannot agree on anything. You will never get a majority vote on it.
As for AV, they would have to have a vote on what type of AV they want and the same SNAFU applies.
This is and will always be the problem, asking MPs to decide when the voting public have already decided. Only a few weeks to the end of recess and the animals return to the jungle to decide on what they do not want but not what they want.