IF you are lucky during your drilling process as you move up in drill bit size, there might be enough heat generate during the process that you won't need to go thru too many sizes before the plug will likely "unfreeze" and you can twist it out.
Especially if you use a left handed drill bit.
Go to Harbor Freight, Amazon, or an industrial supply house and get a set of drill bits with a left handed twist--they work great for this purpose. There's a good chance it comes out while you're drilling it.
Remember to run your drill motor CCW.