Will it be possible to export a key and only allow it to be imported once?
No, there is no way to keep track of what people do with their private keys. You could easily write down the key before exporting it and you can use it without ever importing it to a wallet. It's possible to construct a transaction from the private key manually or using some 3rd party software and then broadcast that transaction to the network.

It sounds like you want to be able to send money without using the blockchain by simply passing private keys around. You then run into the double-spending problem. This is exactly the problem that the blockchain is solving. So, no, if you want to securely send bitcoins you will have to use the blockchain.