Most immunosuppressant drugs block cell-mediated immunity, and so attempt to prevent cytotoxic T-cell killing of the donor lung tissue. Unfortunately the lung tissue will also contain a large number of the donor's immune cells (mast cells, macrophages and any eosinophils that are in the tissues). It is acute triggering of these cells, and release of bronchoconstricing mediators, that results in anaphylaxis. These are relatively long lived cells so the graft-mediated peanut sensitivity will take a long time to decline.
Immunosuppressants *should* block Graft-versus-Host reactions as those are primarily cell-mediated.