Backing fiat money with bitcoin (or gold or anything else) would convert the fiat money into an honest money substitute (and no longer fiat) again. Fiat money is "money by decree". It is in a very real sense fake. The only things which gives it value are the cultural memory of "dollar-ness" coupled with the legal tender laws which make it illegal to issue private competing currency.

Bitcoin has most of the properties necessary to make it money. If and when it is widely considered to be money, then it could conceivably be "kept" in a safe place (e.g. the distributed network in which it currently exists) as backing for some unspecified, more transportable surrogate. In the heyday of private and state banking, the checks cross-written against groups of banks were cleared every day at a clearing house. The differences at the end of the clearing calculations were then settled between the banks by physical gold transfers. This was done continually all over the world in a local, regional, national layered fashion with the final clearing done in London. Only the interbank differences were settled in gold.

The days of the gold standard did not mean that everybody carried gold around in their pockets. It only meant that their "paper money" was fully backed by gold. And "at the end of the day", all accounts were settled. It worked beautifully. This may be a good model to think about with respect to Bitcoins. We don't all have to deal directly with Bitcoins, as long as the accounts are settled in Bitcoin periodically. This leaves us free to invent an honest bitcoin substitute that may be more convenient to use, but is backed by Bitcoin (like gold used to back bank drafts).