I was sued for breach of oral contract in court by a former romantic partner. The judge found her testimony compelling. She had a lawyer, I didn't. The judge ruled that I owe her in excess of $40K.

I was laid off 3 months ago. I rent a one bedroom w/my daughter. I own a 10 year old car. Unemployment runs out in June.

Can I declare BK and name the judgement?

If you file in bankruptcy you MUST list all debts. I don't know why this particular Judgment would be protected and not discharged in Bankruptcy unless child support or spousal support (which is obviously not the case) is involved.