Workarounds for setting default schema for domain group?
Taking @Mr.Brownstone's idea, you could definitely try this:

DECLARE @sql nvarchar(MAX) = N'';

SELECT @sql += N'
  ALTER SCHEMA dbo TRANSFER custom_schema.' + QUOTENAME(name) + ';'
  FROM sys.objects
  WHERE SCHEMA_NAME([schema_id]) = N'custom_schema';

EXEC sys.sp_executesql @sql;


Now, it may be the case that you have foreign keys etc. that will need to be dropped and re-created (and I've shown how to automate that too), or as I mentioned in a comment, you may have explicit references to custom_schema in your existing objects. You may also have duplicate objects (say, dbo.Customers and custom_schema.Customers) that you'll need to handle. But the general idea should be sound.

Another workaround would be to upgrade to SQL Server 2012, where you can set a DEFAULT_SCHEMA for a domain group. I haven't tested this, but this Connect item says it has been made so.

In either case, I still think the app should be fixed eventually. As I've mentioned many times before, you should ALWAYS use an explicit schema prefix when creating or referencing objects.