@simbobs
I think it will be hard enough to get needle phobic DS to have a covid jab when and if one becomes available, never mind the flu jab as well.
DD1 isn't keen but does it, dd2 loves to make a fuss, ds finds needles fascinating... and then normally faints about 30 minutes afterwards, much to dd2's jealousy. 
I gave the school a conundrum last year with ds. He had a jab, so I put that he normally faints, (which means they do the fainters first) but also his temperature normally shoots up about 2 hours after the injection so could they do him last in the hope he'd get a full school day in.
I think that confused them so they did him with the rest of his form. Then sent him to PE the next lesson, so he fainted in the middle of that, and by the time he'd revived his temperature was up.
He was smug. I told the school I'd told them that would happen, and dd2 was seriously unimpressed. 