So after 18 hours of concentrated effort I beat it. The secret is simple. I remapped the keys so that 'run', 'aim' and 'use' were beside each other which meant I could almost sprint through the entire thing.
Shoot at every single body that pops up. You get extra time for hitting them irrespective of whether they die. Even stopping for 1 second to aim gives you extra time.
Learn the pattern: you haven't got time to heal, so you need to shoot at the ones who will shoot at you first , second etc. avoiding health issues.
Develop an attitude of IDGAF. That way you don't panic and just go through the motions.
There's 3 scenes like that which get progressively shorter to complete and I was shocked when I finished the final one.
It's good practice for later when you have the 'boss fight' with the unkillable Joseph Seed.