I have an ASP.NET app that takes multimegabyte file uploads, writes them to disk, and later MSSQL 2008 loads them with BCP.
I would like to move the whole thing to A
You are on the right track. The Bulk Copy API will work. I am using it. And it's the fastest way to import data because it uses INSERT BULK statements. Not to get confused with the BULK INSERT statement which is not supported in SQL Azure. In essence, BCP and the SqlBulkCopy API use the same method.
http://www.solidq.com/sqj/Pages/2011-May-Issue/Migrating-Data-into-Microsofts-Data-Platform-SQL-Azure.aspx for a detailed analysis of options available
I think it's important to note that BCP cannot handle source files that are Unicode while using a format file to do your import.
BCP is one way to do it.
This post explains it in three easy steps: Bulk insert with Azure SQL