AgreedWelcome! I'll agree with everyone else, no need to take off the FEL or BH to mow. Full disclosure, I was nervous when I first got mine in the on/off procedures. I mowed a few times with the extra 1k lbs on. After learning the on/off procedure, never again, not cause its "bad" but because it's too easy to put on/off and mowing with a 15' tractor isn't the easiest.
I don't have a backhoe but the "back plate" you speak of must be the mounting frame. As far as I know it does not have to come off to use a mower as that wouldn't make any sense. And I'd question why your dealer "isn't sure"... he sells these things so he should know.This is my first John Deere, all these easy quick connects are new to me for sure! The reason why I ask this question, is because my dealer wasn’t sure when I remove the backhoe if I have to remove the black plate as well behind the tractor or not. He isn’t sure if the mower deck would fit using quick connect or not. I would take the loader and backhoe off but there is a black plate that the backhoe supposedly hooks to that he said would possibly need to come off?