Yes, they are if they are solo mining (and they technically understand how to do it). If they are mining in a pool, then the pool needs to pick the transaction (which some pools are willing to do, and others are not).
F2Pool has been known to do the exact thing you are asking for (confirm a transaction in exchange for an out-of-band payment) on multiple occasions. I'm not sure how big of a payment they would want for your 225 byte transaction. You could contact them and make an offer and see what they say.
I believe that pool is operated by a bitcointalk.org user that goes by the name of macbook-air. You can try contacting him.
Be careful using abandontransaction. Doing so only removes the transaction from your own wallet. It does not cancel it on the network. Any network peer that has already received the transaction (including, but not limited to the recipient of the transaction) can rebroadcast that transaction anytime they like so long as the transaction inputs are still unspent. If you abandon the transaction and create a new transaction (with an appropriate fee) to send bitcoins to your intended recipient, and your new transaction doesn't use any of the same transaction inputs, then it would be possible for the recipient (or anyone else) to rebroadcast your original transaction (and pay a pool to confirm it). Then your recipient could potentially get paid twice from you.
To avoid this risk, you can take note of the current transaction inputs before you abandon the transaction, and then use the coin-control feature of Bitcoin Core 0.12.1 to choose at least one of those inputs for your new transaction. If you aren't sure that you can figure out how to do that, you could just send your entire wallet balance to a new address after abandoning the transaction. This would combine your entire wallet balance into a single brand new transaction output so that your abandoned transaction would no longer be valid.
Wow, thanks so much for this awesome info! I'm going to contact the user you spoke of and see if he can help me out somehow. As for the abandontransaction command, I actually tried it yesterday but it failed and threw an error code which said: "Transaction not eligible for abandonment (code -5)"
I think the best bet at this point is to try to get a solo miner or pool operator to mine it for a reward. I mean, it's only like $20 worth of BTC, but it's just an unpleasant feeling knowing the funds are floating out there aimlessly in the void.
Thanks again all for the help. If anything, I've learned an important lesson here