I had very similar issues when the weak BEC on my MMM was causing my Rx to brown out. Once I fitted an external BEC all problems went away. I think you might be having problems with the BEC being unable to supply the demand from the servo and the Rx but if possible also try to eliminate the Rx by changing to another one if you can. Either way a decent BEC is only about $20 so it isn't a huge layout. There is also a slight chance that your servo might be causing issues. Sometimes a faulty servo can cause the BEC to go into shutdown if it causes the it to overload. It's just going to be a process of elimination.

Also what brand are your batts? 3S 8000mAh 20C??????