My understanding is as follows:
When you import your private keys it asks you for an account name. If you go to the console you'll see transactions like from GENESIS -> ACCOUNT. If you import the private keys into a different account in a new wallet it may still attempt the transaction but there is nothing to move as the shares are now associated with ACCOUNT.
The BTSX from a reserved Founder ID is different in that the account was created as part of the Genesis block. You'll need to manually move that to a different account. The account that you transferred your other funds to (from the Genesis block) would be fine. However unless your Brain Wallet Passphrase is compromised the Founder ID in the Genesis block is safe.
Account names are like bank account numbers. You can have many of them. If you want to be extra sure create another account called ACCOUNT_NAME-Savings or something and move the bulk of your BTSX there and only transfer BTSX back to your main account when you want to do a transaction. Unnecessary but it might help you sleep at night
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