How should I install a partition wall perpendicular to a finished exterior wall?
If this is just going to be a partition wall, why do you need the header? There will be no load-bearing issue to address. Thus, no need for a header.

If you want an opening through the partition wall and you want to drop the frame down from the ceiling, just use cripple studs, or put a header just over the opening.

I would strongly consider cutting a channel in the sheetrock. attach the top plate of your wall to the joists above, then build the wall directly under that. Again, no load-bearing issue, no need for a big header to bear the non-existent load. ;-)