Is there a way to omit info in a task when moving it from one project to another?
The only thing I can think of is the ability to modify a task name using Flowsana, but this adds to the task name, which is not a solution for you.
It is now possible to change the task description, but you can merely replace it, not change it.
So, no I do not think so.
Not a perfect idea, but one that might help:
Using Asana rules, you could create a new task (in the other project) and removing/completing the first one.
That would be equivalent to moving it, and might help you to get rid of some information.