Secondly I find it hard to reconcile being able to achieve a 7000fpm ROC in two and a half thousand feet from F350? Even if it was possible the ballistics and KE of a 205 ton aircraft going up at that rate would see it going quite a bit higher IMHO. In a light aircraft at lower levels it is extremely rare to see more than 6000fpm with a light aircraft never mind the pitch rate to achieve that ?
I think we're getting a little fixated on the 7,000ft/min climb. That was reported by the BEA as the max rate of climb - we don't actually know if that was sustained for any length of time.
We know at 2h10m05s they were ate FL350 and at 2h10m50 were at FL375. So, that's actually less than a minute for a 2,500ft altitude gain. Overall, the rate must have been much less than 7,000ft/min for a good portion of that climb.
We also know there was nose-up sidestick but updrafts could have played a part too.