Look for your answer in the Philippines
I am Korean staying and working at one of the flying schools in the Philippines. You can have good flight training and able to have chance to work as FI here. Many forgein students are learning to fly from India, Korea, Malaysia. You can finish entire courses in 6-7 months even up to FI course. Within your budgets, it seems difficult to get trained in States cause other than flight training cost you must consider another expenses for living. It will cost you double the budget what you have. Hope this is good information for you.