Thank you for mentioning the isolation level! I never changed it and I don't know what the default is, but because you did I did some google searching. I just learned that bulk copy operations block queries running under
SNAPSHOT, READ COMMITTED, and READ UNCOMMITTED with row versioning isolation levels. Something tells me that the issue was actually with the default being one of these. I changed it to REPEATABLE READ and it can load all 3 documents into the database at once now. I'm having issues with the transaction ignoring the timeout value I set now, though...