How does Tet not recognize the body inside the coffin?

If the Tet could unerringly identify DNA behind layers of rocks, it would have rid the Earth of the Scavs long ago. The reason Beech's bunch wore those Predator-like costumes was to defeat the Tet's scanners. While it would have been nice if the director had made this plain to see, one can surmise that the pod that Jack took up to the Tet was concealed using the same technology as the costumes.

More importantly, the camouflage technology was also used to hide the nuke.

(You could argue that Beech didn't even have to be there. But he did say earlier that he'd really like to see the look on Tet's face when the nuke went off.)

(I agree that it's all rather poorly handled.)

I feel that this part was not answered in the accepted answer:

Given the Tet could recognize DNA behind layers of rocks.

(as shown when Vika requests drones to be sent to rescue Jack.)

Here are the images of the actual sweep:

1. Drone 185 is sweeping a ground, far from where Jack currently is.

   

   If it wasn't far, they wouldn't have the time for their long "starting to reveal what's really going on" conversation.

2. The analysis itself says "bio trail".

   

3. The actual trail, as seen by the droid.

   

   From the legend on the right, the cyan trail is clearly Jack's, while the red one belongs to an unidentified person (I assume Julia's).

   The absence of a third trail confirms what @coleopterist wrote in his answer: the rebels' costumes are there to defeat the Tet's scanners. Otherwise, there'd be at least one more trail, since the rebels seem to do everything themselves, i.e., they don't seem to have drones or similar technology they could use to bring Jack in.