If you pick them up, they're no longer prone
There's no support for that in the rules. It doesn't even logically follow from the rules that are present, as since the rules actually say it takes a move action to stand up from prone, without some other rule input, it logically follows that it is impossible to physically force an opponent to stand from prone, as it can only be done by them using up an action.
And even without that, once again, nothing says a prone creature can't be tripped, RAW.
The FAQ clarified that, when talking about re-tripping with AoOs provoked by standing from prone iirc. Plus, they can only be prone once, but the result of a successful trip is making them prone (or making them fall if airborne). If you can't make them prone (or fall), the trip is
not successful, and Improved Trip won't trigger.
So, even if you could trip a prone target, by RaW, it doesn't matter, you still wouldn't generate extra attacks beyond the first, even if you could get improved grab on ranged attacks with a crescent knife.