Setting the value of the AssignedTo field should be relatively easy, except people are accessed through this confusing concept of "Claims".
Assuming that you drop down is structured exactly as is needed for writing to the AssignedTo, then you're golden. If not, then you'll need to create the claims structure to write it in.
To eliminate all that, maybe you need to do this:
1. Create a security group and add the Managers.
2. Update the AssignedTo column to only select from the newly created group.
3. Show the AssignedTo dropdown on the form, rather than your custom Drop Down.