Private halls aren't going to be cheaper than Imperial's, and you'll miss out on some of the social aspect. Your best bet is the halls further out, like Woodward and Kemp Porter. If your budget is max 200 a week, then you can definitely afford those.
In 2nd year onwards, a lot of people move a bit west; e.g. Fulham, Hammersmith, West Kensington. In those places, if you're sharing a flat with others, then around £150 per week is possible, but this excludes bills a lot of the time.
Alternatively, you could opt to be a hall senior which would allow you to stay in halls, but you'd have to take on a couple extra responsibilities.