It seems like there is nothing I can do to rebroadcast this transaction with a fee, and that I have lost this $1 forever. 
At least it was only $1, though.
At least it was only $1, though.Hmm... Not really. Can't you just remove the private key from bitWallet? Then it can't rebroadcast the transaction at all.
Unfortunately, BitWallet is not the app that is rebroadcasting the transaction. BitWallet only broadcasts the transaction once (according to the BitWallet developers). It is other nodes on the Bitcoin network that are continually rebroadcasting the transaction.