I think a common box addressed to BP intended to collect all business related material is a good compromise. That way if either of you get business related material, it can go in the box.
This way H isn't out of the loop - no secrets or suprises - but he also doesn't have to take responsiblity for sending stuff/taking active involvement in the business. If he really wants to, he can look in the box to see what is there, but he can also symbolically "dump" stuff into the box, and have you help him "dispose" of it (by mailing it to BP, not shredding - don't want you to get carried away now ).
LG, packing material of shredded cards, love notes, etc., I hadn't thought of that - but I've got to say, that would have been a classic. Years from now, when you tell this story, I'd say you did it that way. Adds another level of flavor to the story.