The second problem is of course those very energetic neutrons -- to make them not activate enough material to produce even worse fallout, you need to wrap the bomb in something that absorbs them and produces something benign.
Also, just as a minor nit that always bothers me: Plutonium is never your problem with fallout. If a bomb goes off or a reactor burns, the other things that are released will always be many orders of magnitude more dangerous than the plutonium. No-one has ever died of, and no-one will ever die of cancer because of plutonium released in a nuclear explosion, reactor leak or a reactor failure; simply because to have enough exposure to plutonium for that to be likely, you will have developed a thousand other cancers caused by other, more dangerous isotopes that you would also have been exposed to.
(People might have died cancers they got from working on plutonium for bomb, rtg or other use, but that was only possible because the plutonium was first separated from the more dangerous isotopes and well purified.)