Hi Ben,
I might not have an answer to movement of attributes. I was wondering if there could be a workaround of sorts. Following is what I could think of.
1. For objects where having the attribute does not make sense, one can remove the attributes from the relevant layouts.
2. One can adjust the attribute visibility as well from the attribute details panel on the Type and Attribtue Manager.
On second thoughts, I found case CS64539. This talks about TypeAttributeMoveTool Utility. I think this should help your need.
-Malavika