I want to be able to specify which bitcoin miner receives the fee. Is that even possible?
Not directly. You could do it by transmitting the transaction only to the miner in question, and making sure that his software will transmit the transaction to no one else (note that standard clients don't work this way; they're happy to retransmit transactions quickly across the whole network). This would mean that only this miner is trying to include it in a block, so he's the only one that can get the transaction fee.
Note that this means it might take a prohibitively long time for this "transfer" to go through, since you have to wait for that miner (or pool) to find a block. In the case of a pool, the transaction fee goes to the whole pool, not a certain miner in the pool (unless you have some off-blockchain communication).