Still technically doable without Woesel

If you heavily invest in both Persuasive reducing and increasing gear, it's still doable.

Using Persuasive reducing gear, you can get around -57 (more-or-less) Persuasive. Then lower your base persuasive to around 175 using Ridiculous Hats, allowing you to draw the card while wearing the reduction gear. Once drawn, switch to max Persuasive boosting gear, which should give you around a 90-100% chance to succeed. Works best with SMEN and Fate items, alongside Light Fingers for access to Tatterskin Shawl and Lyon Pursuivant of Arms Extraordinary.

This strategy will become less demanding as better Persuasive increasing and reducing items are added to the game.