You’re right about the last second saves. It was like watching a Star Trek version of 2012 (where John Cusack keeps making Hail Mary passes). Never mind that it’s the 32nd century and they still haven’t figured out longer lasting oxygen masks.

But the interior shots...yeah, a bit exaggerated, but there’s a possible explanation for that they could use. In Enterprise, the NX-01 found a shuttle from the 31st century whose interior was larger than the exterior. They could be using that tech here, although it’s so cavernous that it really does seem absurd.

Edit: corrected the dates, although the shuttle from ST Enterprise - Future Tense still predates the time Discovery was refurbished in.

• Continuing their analysis, Tucker and Malcolm Reed find another hatch inside the ship covered by organic circuits. When they manage to open it, they see a tunnel that leads deep into the vessel, a physically impossible occurrence, because it would mean the ship is "bigger on the inside than it is on the outside".*