Filing a document with a government agency signed by a fake name would certainly appear to be a problem. However, if this is AA's long-standing practice, the DOT most likely doesn't care. A judge might care more, for example, if you list a number of AA misrepresentations, add the name to the list, but first make sure it really is a fake name. A possibly useful litigation phase is "upon information and belief ..."
This is probably more of something that might make them look bad than anything of real consequence.