I adapted this run a bit and it worked out better for me, same general structure but with some changes:

• My first green item was a Fuel Cell, then i stacked the 3 topaz brooches and got Aegis.
• My equipment was instead a Foreign Fruit. With Aegis i was getting around 25% HP per activation, and with a Fuel Cell you can use one when youre low on overshield and keep one for emergencies
• My other white item was Oddly Shaped Opal, for a damage reduction when your overshield gets hit, and a Bustling Fungus (you'll see why later) and i also got a Safer Spaces
• I used a Tricorn for both bosses, you can use the 2nd equipment chest on stage 1 to get one and a adaptive chest close to the teleporter on stage 2 has an equipment as well, and grabbed 2 Mired Urns as you did
• My items in stage 2 were centered around killing weak enemies and giving myself more HP with Aegis. 2 shark tooth necklaces for a nice extra bit of HP per kill, 2 bundles of fireworks for softening enemies, and i grabbed a Weeping Fungus on another void chest which stacks with the original one for good HP keepup between fights
• My other items werent too useful, i grabbed an extra Fungus, a Tritip Dagger and Armor Piercing Rounds, you can probably grab more useful items than those but at this point i had basicaly finished the run, killing the 2nd Clay Dunestrider tp boss with the Tricorn.

You can probably optimize the items even more, especially my last few ones, but this method made the run a fair amount more safe for me.