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Actual solution by Capn SlashD from TrueAchievements:

I would recommend doing this in New Game + and on the Assisted difficulty level, to make it easier for yourself, with the equipment listed below.

Red 9 (Acquired from the destroyed boat in the middle of the lake in Chapter 4)
Powerful handgun, not the best accuracy, but improved with the Red 9 stock traded from the Merchant with spinels. Exclusive modification improves power by 1.5x, making most enemies go down fast.

Primal Knife (Acquired from the Merchant after destroying all 16 Clockwork Castellans)
Although not the most powerful knife, it's exclusive modification is infinite durability, meaning you don't have to worry about it breaking during critical moments. (Also, helpful if combining with a no merchant playthrough)

Silver Case
Improves handgun ammo drop rate.

Optional - Any charm which gives extra handgun ammo from crafting.

The things that WON'T void the achievement according to my and the communities playthroughs:-
- Using a handgun or the knife to attack.
- Equipping another weapon, but not firing it.
- Shooting explosive barrels that then kills an enemy.
- Using the harpoons during the Del Lago boss fight.
- Using the cannons to kill enemies during the castle section.
- Ashley killing enemies with the wrecking ball in Chapter 14.
- Using the turret to kill enemies and destroy the AA gun in Chapter 15.
- Using the Special Rocket Launcher during the Saddler boss fight.

Throwing an egg, hand grenade, flashbang or firing any other weapon WILL void the achievement.

You can check the status of the achievement under challenges on the pause menu, if the challenge "Minimalist" is more than 0, then you have voided the achievement.

I would recommend putting all other weapons into typewriter storage, this will increase the chances of getting handgun ammo, as the game will only spawn ammo inside of smashable boxes/crates of the guns in your case. Just remember to shoot to daze then melee attack and stab or slash enemies with knife when on the floor if no immediate danger around to try and save ammo as much as possible.

If you succeed in making it through with only your trusty handgun and knife, the achievement will pop on the result screen at the end of the game. Good luck!

(Thanks to Frooga, Ov3rExile, xCENAxMOD, Mr Busuto, Cruxgen, Raziel Cruentus, BAK3D B34N, MrRedSoloCup, Maka91 and DanTheWhale for the tips to add to this guide.)