With this said do I really need a better CPU?
The clock rate of your choice is very good. However, in the past AMD processors weren't able to use more than one core in AH. For what I've understood the issue hasn't been totally solved, or at least it's poorly documented which processors work and which don't. In that light Intel might be a safer choice.
As BaldEagl said, even a highly clocked I3 would do perfectly well. It's a dual core one, which is sufficient for both AH2 and AH3. Changing the processor and motherboard in your pcpartpicker list raised the total to ~$840.