You could, if you're willing to risk it, look at some of the resale sites. www.kelbillet.com for example, which has 5 tickets for sale on that date.

(Strictly speaking, this is not permitted, but Eurostar do appear to turn a blind eye to it - hence the existence of these sites. I've travelled this way a few times - even with a ticket for the opposite sex...)

Alternatively, there are options via Lille (as noted above - and a good place to break your journey for a decent meal) - and also Disneyland, an easy RER ride from Paris, currently selling for £74.50 on the 6th, if you can bear the number of children on board

You could also go old school - TGV to Calais, ferry to Dover, high-speed train to London. It's quite enjoyable, but does take a fair bit longer.