Stop trying to achieve the challenge : losing 4 times means losing 21 dices. Take your best deck with your best character. I used Mako with a Hammershark deck. I won at my first try.
If you're that desperate, you could always try Armageddon Designator + Relinquished. Obviously not a terribly great solution, but Relinquished IS basically the only threat in his deck (though you have to be careful of Golden Apples), and Armageddon Designator permanently neuters it if it works. Otherwise, Ritual Sealing and Ready for Intercepting and pray.
I used ritual sealing to banish Relinquished and he used his special skill and called him right back next round. I just wanted Yugi Muto so I beat his level 20 instead. I will aim for 40 the next time threw.
Ritual Sealing, Skull Lair, The Emperor's Holiday, Ready for Intercepting, Possessed Dark Soul, or untargetable things like Hazy Flame, Chaos Command Magician, and Toon Kingdom. Use Destiny Draw skill to draw them
Use the mighty Sacred Phoenix of Nepthys!!! Works on both Pegasus beatifully. After destroying his backrow over and over I stall until I can Ritual Summon Magician of Chaos. I'm using Destiny Draw (beacause is Yugi against Pegasus afterall!!); balance should work too
1. use dimension gate to one of your monsters with high atk (at least 1600-1700). after that he won't attack and 🔥 him through dead draw
2. use dna surgery and Buster Blader, the Dragon Destroyer Swordsman to stop him using all of his monsters effect and so cockblock him (should also work on pegasus 1)
Easiest Way -- completely fill your deck with lvl 4 or under monsters -- always summon them in defense mode -- Reliquished won't attack and Pegasus will deck out