Is paying a bribe in this case considered a matter of repelling a greater evil with a lesser evil, or does it cause one to incur the curse mentioned in the Hadith?

Bribery is what one uses to uphold falsehood or abolish truth. What you mentioned is not considered bribery, especially if what you did was after your repentance from the seclusion (khalwa). The fiancée is a stranger to you until the marriage contract is concluded, so seclusion with her is not permissible. Her father should hasten the marriage contract.