
I would say that absolutely, this could be being made worse by her season. I'd keep him away from children altogether until she's fully over it and a few weeks on top.
As far as the children go: no, I would absolutely NOT muzzle him and let them play around him. He is not totally happy about them now, doing that will only make things worse. Whilst muzzling might be a sensible precaution, it can also lull people into a sense of complacency because they know he can't hurt them, and they can then inadvertently get too close and stress him out, making the problem worse in the long term.
You need to back right off, and take this VERY slowly. Treat it as you would a phobia in people: facing your fear very rarely works because it's too much too fast; what works is gradual desensitisation and counterconditioning at the person's pace, and the same is true of dogs. You need to work at a level that he is comfortable with - this is what's known as keeping him 'sub threshold', i.e. working below the point at which he cannot cope and begins to growl/shows stressed body language etc. So, for example, if he begins to become uncomfortable with a child at 20 feet away, work at 25 feet. It's great that he is choosing to take himself away - that sort of decision is exactly the sort you want to encourage, because it is an active decision to avoid confrontation or aggression. Great!
Desensitisation is basically getting him used to children - the mechanics of the ball game you describe is a good way of doing that. Very little pressure, and rewarding too. Counterconditioning is changing the associations he has with children - I find that food is by far and away the fastest and most effective way to do this, because eating food releases endorphins and triggers dopamine/serotonin production, both of which make him feel better. So, pair children with food! A good way to start is the open bar/closed bar method - you are the bar (the source of food), and when a child appears, you start to feed him a constant stream of small, really tasty treats and when the child goes away, you stop. So child appears, bar opens, child disappears, bar closes. It's a very good way to make a very strong positive association quickly.
At some point in this process he should start to look to you when a child appears for his food. This is good as it shows he is making the right connection (that a child is a predictor of good things happening, not scary things). So, now you can start to reward him for looking at you when he sees a child. This will continue building the positive association, but also start to build his responsiveness to you when they are around and also his awareness generally (so rather than just focusing on the child in a fearful way, he starts to think about what might happen to him in a good way and becomes more aware of what he's doing that's getting him a reward).
From here you can start to reward particular behaviours. Reward literally anything that is not a growl (as long as he is at a comfortable distance, remember): looking, turning or walking away, sitting, lying down, doing nothing. Nothing is actually a very good behaviour to reward! Sounds a bit odd but it's underrated as a behaviour - if he's doing nothing then he's not growling, so it's good. So if a child walks past and he does nothing, reward it!
You are already doing a stellar job with management by stopping the kids touching him or being too close, that is brilliant :-) You need to maintain that out and about if any children want to touch him - be very firm, do not let them. Any time he gets unwanted contact will set him back.
On that note I'd also stop having him around your friend's children for the time being while you work on this - if his legs are shaking when she's near then he is already over threshold, and learning will be severely impeded. He needs to be totally comfortable and calm for you to have any chance of success. If you are visiting her at her house, leave him at home; if she is visiting you at yours, put him away somewhere quiet with something to do (treat ball, stuffed kong, that sort of thing) so he's busy and feels safe.