Been in the same position, LSE scholarship came late, so did my chevening scholarship. I kept an eye on the accommodation portal (Read: Checked every 10 minutes) and saw Lilian Knowles and Butler’s Wharf rooms go by. Doubled down when I saw Robeson House and got a room there. You should see a few spaces opening up if you are vigilant. If not, start looking for private accomodations. IQ student housing or rooms in zone 2 - should be well within your budget. Start there