GPU is usually faster, but I cannot find GPU benchmarks for Handbrake, so I'll recommend a CPU. Keep in mind that using the CPU I recommend, you need a new motherboard.
I'd recommend an AMD FX-8320E. The 8320 because it's practically AMD's top of the line, and the 'E' because that basically means it's better binned, and they are both the same price at the moment.
Other popular AMD CPUs such as the 8350, 8370, 9370, and 9590 are basically overclocked versions of this, but I can't bother to provide a link, so you'll have to search it up or not believe me.
I recommend AMD, also, because of better price/performance ratio. As you can see in this anandtech benchmark, the overclocked FX-8320E ($125) is on par with the non-overclocked i5 4690 ($200). Overclocking Intel CPUs would probably surpass performance, but not by anything worth $200, which is already at your budget, so you wouldn't be able to buy a new motherboard.
I know that some people are going to point out that it is hard to upgrade an FX-8320E to anything, due to an outdated chipset, but I don't think that 'dark fang' is going to be upgrading any time in the near future.
Unfortunately, when using an AMD CPU, you are going to have the disadvantage of not having a built it GPU. You can either get a motherboard with on-board video, or a cheap dedicated GPU.
I'd recommend a motherboard with a built-in GPU, which I know everyone thinks is terrible, but looked at his current iGPU. A nice, overclocking motherboard such as this, would be ideal for getting the most performance out of your CPU as well as getting an integrated GPU, but a more budget-friendly option could be this instead. I know option #2 isn't the greatest, but it is cheap. If anyone else has a motherboard suggestion, comment and I'll edit into my answer.