The most important part about this achievement: the timer starts the moment you gain control of your character when Turner pulls you from the water, and continues until the bangalore is successfully deployed at the seawall.
It takes 40 seconds from original control of the protagonist to the checkpoint after picking up the bangalore, and it takes 25 seconds for the cutscene and bangalore deployment at the seawall.
This means that, from the checkpoint after picking up the banger, you only have 55 seconds to get across the beach.
Apparently my run in the first video wasn't fast enough, so I shaved 25 seconds off my last route to get up the beach within the new time limit.
The key is waiting in each shown cover area about 3 seconds for the machine guns to "forget" you're there.
Skip to
2:24 for the beginning of the route.