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So, this is the strategy and equipment I used and works under the assumption that you are at lv85 and have trained health and mana potions to max level. (This has worked 3 times in a row)

Armour: Paladin 

Weapon: Spystaff of Static +15 Nature 
Shown: Blade of Awe (Has a 5% rate to heal 4% of health and mana)

Accessories (all non-rare and non-seasonal):

Aika Backguard +10 Nature + 2 All
Grimey Ring +10 Nature 
Neutral Mask +12 Nature
Cheesemonger’s gratitude + 12 Nature
Man’s best friend + 5 Nature
Leorilla Mane belt IV + 10 Nature
Legion Bracer +1 All

TOTAL 

+74 Nature resistance + 3 All

Character stats: 
200 END 100 WIS 120 LUK

Other relevant gear:  
Cloak scrap (to farm for 5 potions)  Seaweed and Rotten Hardtack as backup

Strategy One:

Leorilla:

Holy storm (stun, 12), Holy wrath (10), Phoenix song (2) , Cor Lucem (6), Crusading strike (1), Phoenix song (2).   Keep repeating Crusading strike and Phoenix song until either Cor Lucem, Holy wrath or Holy storm becomes available again. Also, use healing potions, angelic blessing or Greater heal when necessary (e.g. as soon as angelic blessing and greater heal is available and healing potions when health is approximately 1000) Leorilla should be defeated at approximately turn 60 and around 2-3 potions should have been consumed. If  shielding is attempted it is to be done with care to avoid enraging Leorilla, and less potions would be consumed. Use swords of eternity as a filler if necessary.

Sir Bearginion:

Barrier of Sanctuary (+140 MPM for 2 turns, 2), Prismatic strike (-20% boost and -25 light resistance for 4 turns, 4), Phoenix song (2), Light of Dawn (Reduce bonus by 25 for 5 turns, 4).

Following light of dawn, fill in turns with multi-hit attacks to trigger BoA’s special until barrier of sanctuary can be used, then rinse and repeat. At this point, between using your shield, debuffing and using angelic blessing and greater heal, you should be able to outheal Sir Bearginion’s attacks and should not have to use the remaining two potions or the backup seaweed and hardtack. Though, mana maybe a problem so use either mana boost or a mana potion if necessary in between rotations. Sir Bearginion should be defeated around turn 138.

Notes: 

Other people that have defeated the duo with Paladin have defeated Sir Bearginion first and then equipping a pet with high damage pet to defeat Leorilla due to the low DPS of Paladin (or a healing pet).

Strategy two: 

Sir Bearginion:

Same as main cycle for defeating Leorilla except use it on Sir Bearginion. Use potions, Angelic blessing or greater heal if necessary. Approximately 3 potions are consumed before the defeat of Sir Bearginion. Sir Bearginion is defeated around turn 72.

Leorilla:

Once Sir Bearginion is down, equip a healing pet such as Necsnowmancer, Cheshire twilly or a high damage pet to help take it down faster. Essentially at this point between the healing pet or paladin’s own HoT should be able to keep up with the damage Leorilla dishes out. 

Mainly used: Holy storm, Holy wrath, Phoenix song, Cor Lucem. Fill in cool downs with swords of eternity. Rinse and repeat. Use mana potions if necessary. Leorilla should be down at around turn 114. 

Conclusions: 

The second strategy defeats the duo approximately 24 turns earlier. This would also be more efficient if I used shielding, but, this proves that this challenge can be done without debuffs or buffs.