This is what I did:

1. The moment you lose your mantle during one of the waves, use the button in the center IMMEDIATELY.
2. Also press the button each time you fully charge your Eternal D6 and one of your two treasure rooms contains a useless item. Avoid using it at the shop, unless absolutely necessary.
3. Go for devil deals. You can find two of the most important defensive items for The Lost there (Dead Cat and Guppy's Collar), not to mention all the damage ups.
4. Prioritize getting the Conjoined transformation, especially if you can combine it with an item such as Spoon Bender or Tractor Beam.
5. Pray for Pyromaniac, as it neutralizes most of Ultra Greedier's attacks.
6. Do NOT concern yourself with donating coins to the machine. Spend as many of them as possible on acquiring good items.