Here is my post, minus the last two sentences. I don't have a price cap because I am working, although I do want to save money wherever possible. I have an Avermedia LGP if that will help me stream / record, although I use Xsplit Broadcaster to do that already.
Please append everything to this post instead of having people jump through places to try and help you. Make it easy for us to help you, not more troublesome as people will be less likely to help.
Anyway you seem to have everything you need already. Your specs are good though you might want to overclock your CPU if you have a third-party cooler on it and gain some performance.
You have an external capture device, which is good since it won't affect your performance, but you could also use Shadowplay to record or stream your stuff with very little impact.
you could do a skylake instead