Try giving Bhutan a shot! It is really worth it and also fits in your budget. Visit Phuentsholing(if going by road from Siliguri), Paro and Thimphu. As you have only 5 days, try for flights to Paro, but if you cannot, then do the following:
Paro -1 night
Phuentsholing - 2 nights(not consecutive)
Thimphu - 1 night
Siliguri - Phuentsholing - Thimphu - Paro - Phuentsholing- Siliguri
All the 3 countries offer a different theme for travel. Bhutan is a mountain lovers paradise, Istanbul is famous for its culture and architecture while Sri Lanka is famous for its beaches and wildlife parks. In your budget, toy can easily cover any of the places. So, it's your call, where you want to go.
I thought you go for the Sri Lanka which is really amazing place for Sri Lanka you have to prefer you go through ocean way rather than the plane you just feel the true beauty of your traveling
good luck for your trip 👍 👍