Hello Orm....hmmm...tricky one. I think you are right, he is going to need coping strategies (and so is she, but sadly you can't help her work on hers
).
I've only got dds, not sure I can help much, but I suppose I would try and support his relationship with Mrs B by eg:
a) finding things for him to take in to show her/tell her so he gets to engage with her teachy side , 'makes friends' with her. She might find it easier, as might he, if she gets to connect with him over something.
b) by trying, in conversations with ds, to re-frame her less pleasant characteristics in ways that show you are on HIS side because you understand, but at the same time humanise her - make her less scary (eg use words like 'bit grumpy', 'bit stressed', 'maybe she's finding it hard because'). dd1 had a temp teacher once who was clearly quite angry and hard, not dd's cup of tea at all: we had long bedtime talks about Miss X and how she was a bit stressy, maybe didn't understand stuff about dd, etc etc. By the end of term dd felt quite sorry for her and wasn't phased by her at all - possibly very inappropriate on my part, but made dd feel and cope better!
c) If anything escalates, go in and talk to her - sympathise with/listen to/understand her as much as possible, and then put ds's case.
d) ask his current teachers, who 'get' him, to please explain him to Mrs B! And if you really trust them, explain your concerns to them, in confidence - they might be able to help a bit??
All that said, its really her responsibility to get some support or whatever she needs to help her be as good with boys as she is with girls. Or go teach in an all girls' school 
Sounds frustrating - silly Mrs B!