Just to update this thread on the final outcome:
We received 625USD per person from Singapore and 80% of the cost of the flight in mileage (which was how it was paid) from United for downgrade compensation. Singapore worked with United on that - I did not have to deal with them at all. Don't know what sort of deal was made between the two airlines to accomplish that. United, in typical United fashion, initially tried to simply refund the difference between first and business, but Singapore was reminded of their legal obligations (at which point they worked it out with United).
I know another poster had mentioned that it is the operating carrier that is responsible for payment (in this case, LH - the carrier that ultimately ended up transporting us to our final destination due to Singapore's faulty aircraft) but I don't interpret the law to mean that. It was definitely SQ's responsibility to pay (which is further supported by the fact that they did).
Somewhat unrelated to this EU claim, I also finally received all of my insurance claim money from Chase after months (for the missed, non-refundable hotel reservation, hotel in the airport, airport dinner, etc.). Additionally, due to Lufthansa leaving all of our luggage in Frankfurt (and not delivering it for over a week), they reimbursed me for the new luggage I had to quickly purchase in order to accommodate another trip shortly after this one ended.
While I wish it all hadn't taken so long, I'm definitely satisfied with the outcome.