The early part of this fight plays out like the normal version of this battle. Having upgraded bow abilities and damage can be very helpful in culling down the Bulwark's reinforcements before they get close to you.
Be sure to pull out your shield and walk forward to destroy the yellow and black orbs he summons, as they explode when they hit your shield.
As with most of the bosses in Death's Gambit, you have to memorize the Bulwark's wonky hit boxes and use them to your advantage. For example you can stand inside the Bulwark when he calls down more orbs and avoid taking damage, while still hitting him with your melee weapon or bow.
When the blue spirit comes out, get in as many strikes as you can, then run to the other side to avoid the attack that hits most of the screen. This takes practice to get right so you can quickly return and resume the attack.
Because of the tight margins for movement here, a weapon that doesn't push you forward is actually better. That makes this one of the few instances where the Acolyte Of Death's starting scythe isn't as helpful.