The OP's "which start signal?" question is crucial to answering this question correctly.
5.2.4 says "During the course of fire, after the start signal..."
The question is, on a multi-string COF, what does "the start signal" mean in the context of rule 5.2.4?
Does it mean
A) "the first start signal of the COF" or does it mean
B) "the start signal of the current string the competitor is shooting"?
If it's A, then yes the competitor should be allowed to pick up the magazine they dropped during string 1 during string 2, and the RO interefered with them by picking up the magazine, so it's a re-shoot.
However, if it's B, then the moment the start signal for string 2 goes off, the competitor isn't allowed to pick up the magazine if they dropped it during string 1. They would have to pick it up and return it to their belt before starting string 2.