Much though I am loathe to contradict someone who does the job, my train this morning was formed of 377601 and the first class section (inner end of one driving car) did have guard's facilities at the vestibule contained within the first class area. The guard used it at Coulsdon South, during his extensive and effective clear-out of people in first who had standard tickets. I was depressed by the attitudes of those he removed - a combination of expectation of being moved but taking the chance anyway, and annoyance at being moved as if it was their right to be there!
This guard was also conscientious in his duties as he reported (by 'phone) a defective door (yet another one) between the standard and first sections.
I've not done many 377/6's, and have had no official training, just been told they are no different from normal 377's, so off you go. If you work from whats called the intermediate cab (doing 10 bell dispatch), you do get a chance to sort out the first class areas next to you. I've seen the odd one or two come in, see me standing a few feet away and change their minds. On busy trains you might not get very far, but you do get to check first class.
Many you ask to leave go 'but there are no seats....try the next coaches, lots of seats. You also get the odd 'little old lady' who is a bit 'confused'.