Upgrade your rooms

I started a new game and did the trick of running around in gray mode to build science points and research a bunch of stuff. This enabled me to upgrade the transporter before the Phage mission, which raised the chance of beaming the Viidians from 50% + 35% to 50% + 50%. If I had upgraded either the science lab or sick bay, I would have done the same for the second challenge, which succeeded anyway, but I've had it fail at 85%. I did not realize this would occur, but going forward I will always prioritize upgrading rooms over building new ones. It's more energy and crew efficient and will give better results on these shipboard challenges and extracting resources from POI. The main bottleneck early is tritanium, which is why I hadn't already upgraded all 3 rooms.

If I start a new game without the gray mode exploit, which is time-consuming, I would prioritize researching one of these upgrades in the tech tree. The transporter is a good choice because it also increases your bonuses on away missions.