Am I entitled to stipulate the return of the house that I registered in my wife's name in exchange for her khul', especially given that registering it was a condition for her approval of my second marriage?

There is no objection to stipulating the return of the house in exchange for khul‘, due to the Almighty’s saying: "Then there is no sin upon either of them concerning what she gives up [to get free] thereby." A woman, Thabit ibn Qays's wife, sought khul‘ from him in exchange for returning his garden to him. Khul‘ is permissible if there is a compelling reason for it; otherwise, it is forbidden for the woman to request it. If khul‘ occurs, the woman gains control over herself thereby, and her rights against the husband and the husband's rights against her are forfeited. The husband may not take her back except through a new marriage contract.