DL might check your bags all the way through from DCA for you, but they are under no obligation to do so. But if they do, it'll be based on their baggage allowance, not BR business class. Even if you had a single ticket all the way to SIN, still from the US, the marketing carrier of the first segment determines your baggage allowance, so you still would be on the much lower DL one, but you'd at least have a free one as opposed to paying for it on a domestic itinerary.

I think your only real hope at getting C allowance bags for the whole trip is to keep checking if UA has DCA/IAD-NYC available (but of course they don't fly to JFK anymore), or look for NH/UA IAD-NRT instead of JFK-TPE.