I believe MM was focused on the criminal charges and probably didn't feel his actions on behalf of CD constituted representation in the context of the proceedings. Of course, that was wrong and representation is representation and he showed himself to be less than fully prepared in that moment. With regard to the prosecutor raising the issue, it is common for one side or the other to bring an apparent conflict to the attention of the court even though the only persons who have standing on the matter are the client(s) and attorneys directly involved. Putting the matter on the record forces the court to deal with it as you saw today. Otherwise, it would be possible for a defendant to try and raise the matter of conflict as part of an appeal and/or claim of inadequate representation. So, the court will want to get this handled soon and get both LVD and CD on the record as being in agreement that MM can continue to represent LVD without regard to his prior represetnation.
With regard to this question of conflict of interest on the part of defense counsel, the prosecutor does not have standing to insist on anything other than raising the issue for the court. MM has an ethical obligation to avoid conflicts of interest but in a case like this he can easily claim that whether or not he represented CD in the past does not matter as long as he is solely working for LVD now. Interestingly, either Chad or Lori could raise the issue if they wished and claim that the prior conflict prejudices them by virtue of his having received confidential information that might be prejudicial to them at some point. In that instance, either of them could ask the court to remove MM from the proceeding and the court very well might agree. That likely won't happen because Chad and Lori seem to have a common interest for now.
At the point where capital murder charges are finally filed, there should be a number of experienced attorneys willing to take on the case for the publicity and prestige of handling such a high profile case. Most likely MM will try to hang on as local counsel (read errand boy) but will likely get squeezed out IMHO.