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What happens when they disagree about which transaction happened first? Majority rule? Who decides what the majority is, and can it change if 4 of the five nodes leave the network and are replaced by another 5 nodes?
And if I know that I'm going to create a large transaction, can I do some work precomputing node IDs such that the transaction (which I haven't yet sent out) will hash to nodes that I control? If I control all the nodes storing the transaction, then I can just answer "yes, absolutely, that transaction is valid and hasn't been double-spent..."
The brilliant insight behind bitcoin is the distributed timestamping mechanism; everybody agrees on an order of transactions. I don't see how your scheme solves that problem.