If you have a 'sports affiliation' - loosely defined but it asks for ID - a Hilton MVP rate can be had at the Millenium Hilton for $199/night (10 minutes' walk away). See stay.hilton.com/mvp

You can knock an extra $20 off (before tax) per night if you have AAA membership (this includes the AA in the UK) or if you have a corporate chain-wide discount.

Travelbag can do the four nights for £897 (plus you may be able to get cashback). This wouldn't qualify for a best rate guarantee because of the onerous cancellation conditions - plus you wouldn't be entitled to any status benefits and potentially there's no free wifi, so you'd have to factor in the flexibility (including the chance that the rate will go down), plus the value of the points, breakfast(?) and wifi, and the relatively small chance of a meaningful upgrade. As a diamond I'd go with Hilton on this, but it's a close-run thing.

(Don't forget that Hilton offers a kind of cashback if you book through the avios store - not the BA one - no idea how reliable it is but that's another couple of thousand avios).