This can be achieved in a custom multiplayer match.
Make the following changes in setting:
Game mode - Free For All - 14 Player
Weapon substitutions:
Boltok - Gnasher
Boomshot - Gnasher
Enforcer - Lancer
Frag Grenade - Lancer
Incendiary Grenade - Snub
Longshot - Snub
Torque Bow - Snub
Now get a kill with each weapon and the achievement should unlock.
Here’s a video guide for reference:
Gears 5 - Resourceful Achievement Guide - Defeated opponent with every Pickup Weapon