I think MMA fighters don't like it because it does burn you out (When you are not used to training within Crossfit protocals". If you Crossfit prior to sparring you will not be as strong or quick, which in turn will will hinder your performance. Fighter I know want to be on all the time so as not to get beat down while sparring (EGO). If they would just learn to accept getting beat up a little in training and fight behind the power curve for a couple of weeks they will see the results.
I am not a fighter but I train with alot of the TUF guys, this is just what I have seen and been told.
I think if your ego can handle it, to fight at a weeker state for 3 or 4 weeks would be a good thing, and then as you start to taper off what will the results be then?