Shouldn't be any difference in accuracy. It all depends on the quality of the parts. For example, my Bushy Varminter (.223/5.56) shoots about the same as my DPMS AP4 (.308).
As far as building a .243 from scratch, you will be hard pressed to do it for under $1500. There are not enough companies out there making parts for AR-10's and AR-10 variants. My suggestion would be to order your gun the way you want or purchase one, replace the parts that you want to, and sell the factory parts you replace. I think you could get that done under $1500.
Check out Lucas Tactical. I ordered my AP4 through them. I was happy with their service and they will build to suit your needs.