SQL Server bulk transfer help needed
I can't give the credit to Sir Swears-A-lot because his response was in a comment, but he was right.  My assumption that dropping the primary key and creating a new one would be effectively the same or more expensive than loading up an empty new table was incorrect.

Setting recovery mode simple, dropping the primary key, recreating it (forcing the reshuffle), and adding all my new indexes afterwards clocked in at 40 minutes.  That beat any bcp/SSIS method I'd tried by more than a half hour, so that was a big improvement.

And the transaction log grew by 10 gig but didn't go nuts like some of my other experiments.