Well, culture is won by having more tourism than the enemy has culture, so boosting culture, particularly if they are barely influential with you, could slide them back to popular and prevent victory. Alternatively, the AI does a decent job of setting up theming bonuses, so throwing some works of art at Washington could help him resist a little more.
Reducing the enemy culture is a bit more direct, but more dangerous to your diplo victory, given his influence over other civs at this point (better to nip it in the bud and not let everyone end up the same ideology). Declare war, take his capital which no doubt has lots of tourism, and watch his tourism compared to enemy culture drop like a rock.
If the worlds fair is still available, and may come up before you win, and if you are certain you (or Washington) can win it, the culture boost from that could stave off a victory for a while. If you control enough CS's, you could maybe push though Freedom as the world Ideology to help protect Washington too (if he flips to order, it's likely game over, if you're worried about racing to the finish).
If Washington has culture maxed out, conquest isn't an option, and proposals aren't going to have enough time (or your culture is just too far gone), it might be too late. You could try to do a few little things, like bribe others to war him if there is the danger of him producing even more tourism. The AI rarely builds wonders at war (and seems to rarely build anything other than military units/buildings). If you pop another SP, maybe you could somehow boost culture with it. Even getting +1 culture from your great works may tip the scales a bit if it's close.
Unfortunately, there is precious little you can do to help Washington hold the tide if he's the only one really resisting at this point. Conquest is your best bet, if you can muster the forces, and seeing their capital with all their wonders and great works lie in rubble would be an amazing extra perk.
Those are the only options I can think of.
