to get a UA/SQ combo, this is essentially what you’ll have to do. Even if UA would book it on their website, it would likely be cost prohibitive unless you had planned to book in full fare Y - most SQ fare classes are restricted to SQ ticket stock - that’s an SQ restriction.

Note that while a TA can likely book a UA/SQ combo, it will almost certainly be booked on SQ stock, and therefore not earn PQD. Ticketing convention is it needs to be ticketed on the first long-haul carrier. Theoretically, a TA could choose to book it on UA stock, but they are liable to consequences for doing that, and also you’d still run into the issue of not being able to get discount fares on SQ using 016 stock.

In essence, unless you are willing to pay full fare, very unlikely you’ll be able to both fly SQ and receive PQD.