Interesting question.
With the rules as written, I think I could probably argue both sides....
1. That it is still during a dribble so the ball remains in the BC and, as a result, the dribbler can still pick up the ball and the 10 count continues.
or
2. That "during" a dribble is referring to it being actively dribbled, not during an the interruption. The interruption is more of a suspension of the dribble, not having ended but not "during" it either.
I think that what it boils down to is that these rules were never written with this set of events in mind and anything we decide is really an opinion of interpretation rather than anything concrete.
Now, what would I do? Hmmm. I'm not sure. Maybe I'll just call a travel.
