SSIS data transfers between two remote servers
Yes there will be traffic to Server A during stage 2. This is because the data that has been staged will need to come back to the place that is doing the work (Server A) so that the package can send the data to the target (Server C).

I think you may be doing this the wrong way around to be honest. If Server A has the production database on it then you want to put as little extra load on it as possible. Because of this I would suggest placing your SSIS package on Server B which then only has to pull data from Server A - meaning that there will be no additional processing load on Server A other than satisfying the query that is asked of it.

Then, once the staging has been complete stage 2 can begin which will only consume resources on Server B and Server C. This is a much better approach - I don't see any need to have multiple packages unless you want to split the logic of each stage into their own modules so that they can be maintained separate from one another.

The other benefit to this approach is that you can make alterations to the package and the staging database without having to change anything on your production server - this decoupling allows better separation of duties and will make future maintenance easier for you.

I hope this helps you.