This is only a solution or more likely a workaround for those their achievement is bugged or the tracker isn't working (credits to Powerpyx)
1-Play in offline mode and reroll a new Adventure mode in Yaesha, making sure that The Red Throne is your starting location.
2- Go straight ahead into the building and open the gates to talk to the Queen. Make sure you get her angry, the elites will attack, let them kill you to be thrown in the slammer.
3- Leave the slammer by talking to the guard and get upstairs. Continue going until you’re able to climb a ledge into a room slightly above the floor you’re on. This is the path outside the prison anyway.
4- Collect the Broadsword.
5- Return to Ward 13.
Do Step 1-5 five more times.
Once you’ve picked up the Melee Weapon a fifth time, the achievement may unlock. If not, go back to Ward 13, buy as many Melee Weapons at Brabus (shooting range) or have Ava craft whatever she can. Now the achievement should unlock.
I tried this method and the achievement unlocked for me.