This is a hard question to answer because the arguments are so few. It seems to me that it boils down to just two. I will try to counter both.
1. MPs will waste time dealing with their Constituency Labour Party and its branches when they should be out fighting the Tories.
Answer: Is it such a bad thing to visit their constituencies? I know there are some Labour MPs who try to avoid any interaction with their local party members, but if they want to have good election results surely taking the local party activists with them is a good idea. These are the people who for no pay go out leafleting and arguing for the party’s policies in all weathers.
2. MPs represent all of their constituents not just the local party members.
Answer: This would be a good argument for the MP to stand as an independent or form a new party. If they don’t want to represent their local party members, those are the obvious choices. Perhaps they look at the outcome for some of their friends in Change UK, or before that in the SDP, and think that it may be a step too far away from their career path.