Thanks for the input, everyone. I decided to just give it a try using only tack flux and no extra solder paste, as shown in the following video, with the default lead-free solder profile set on the reflow oven:
I verified the BGA soldered correctly with the trick featured in the video, testing for the expected diodes across the digital and analog inputs of the BGA and making sure there are no shorts between the various power pins and ground.
All in all, that went a lot smoother than expected.