I just finished a run without the shotgun, just to see how far I could get before I got into trouble and I actually had no real problem beating the game. I took my time and didn't push for speed and finished at around 1:20:00.

Pros:

You skip the shotgun, all the ammo and the gunpowders to make more ammo.

Cons:

Your handgun shots have to be precise. I used an infinite handgun to calculate how many bullets I would need for the whole run. It will probably vary and I still need to see exactly how many rounds I should pick up and which ones.

If FPS gets capped, you're very screwed on bosses, so Flamethrower is going to have to be the alternative.

Items you'll still have to grab: Heals Handgun Ammo Grenades Flamethrower? (i skipped it)

As far as I know, only 1 FB is required to get past Mr.X in the jail cell.

You can kill G2 with just a knife and 1 grenade, try it.

You can also skip all the water creatures without using a g/r/b if you time it right/get lucky.

Overall, this is a risky strat but I believe with optimization you can get sub 60 with it and maybe a WR but it depends on a ton of RNG and also unlocked FPS.