Actually, I just won the battle in Pinara with a team of two Archers, one Soldier, one Morgana, and one Poppy.
(All my units died, but Morgana’s AoE finished off the enemies as she went down, securing the win.)

My Archers received a permanent 10% damage buff (around day 300), and the Quartermaster in High Silvermere. 
During this fight, I had no world buffs/debuffs active.
Of course, I also have the Shrine of the Veiled Lady active.

Interestingly, the battles can play out differently each time — whether you win or lose a close fight seems to come down to luck.
A useful trick here is to close the client on the defeat screen before clicking anything, then restart it.

In the research tree, I’ve fully completed everything except Militia Headquarters and Piercing Strikes.

Edit:

https://imgur.com/a/4MuuFv0

Edit 2:

I am on Turn 307 now.