Hi
Imperial accomodation is guaranteed for the first year, and so- due to a very large intake in recent years, I very much doubt that they would provide accomodation for non-Imperial students.
As for the cost- When you are applying, you have the option to specify whether the price-band or room type is more important to you- therefore you are more likely to get accomodation that is the same price as the types that you apply for- when choosing your five preferences, obviously go for those that you can afford! However, this is not a guarantee, as I know many people who applied for all single rooms this year, and put room-type as their preference, only to end up in shared rooms. You don't have to take what they offer you- but tbh you aren't going to find anywhere cheaper in the middle of central London.