There is a way to lock a financial dimension to Site which appears to override all other financial dimensions. Even when a financial dimension is specified on a line the posting gets the financial dimension. If you create this lock there is no way to override it that I have been able to find.
To enable Site-to-dimension link: InvWhseMgmt>Setup>Posting>Dimension link
To set up dimension-to-site reference: InvWhseMgmt>Inventory breakdown>Sites
The posting will always take the dimension from the site the transaction happens at. If you sell an item and it is inventoried at division 06 then the posting will carry the dimension "06" despite anything done to the financial dimensions on the vendor, project, item, or line.