The old SAT has only one more appropiate testing date - 5th December in the UK - correct me if I am wrong. I achieved 1950 on my first go. I am looking to get 2250+ - Harvard, Yale, Princeton Wharton, Stanford, UChicago. Do you think that I can easily increase my score by 300+ within 4 weeks?
Not easily. Practice, practice, practice.
So possible in 4 weeks?
Well, how much of that 4 weeks can you really spend practicing? How much time in that 4 weeks can you devote solely to SAT prep? It is my understanding that you probably have many other priorities that relate to school and you also probably do things outside of school that you enjoy, so it is up to you how efficiently you can use that time. Is it possible? Yes, but only if you can study effectively using whatever time you have left.