(an answer to a question asked here.)
1. Extra security.
2. Whenever artifacts are taken out of storage, you have to go through extensive restoration and preservation.
3. Remember the last scene in "Raiders of the Lost Ark," where the ark of the convenant is carefully packed away in a crate, then taken painstakingly to a giant warehouse with carefully monitored conditions and filed away? My impression is that, to a greater or lesser degree, is a decent representation of what the Smithsonian archives looks like. Getting a bunch of stuff from that warehouse to wherever the filmmakers are filming and back without damaging said stuff probably is quite expensive.
4. The $2,000 per day figure is probably in truth somewhat arbitrary, but documentary filming access is usually bought on a "per day" basis, so the Smithsonian is just going with the industry standard.