Which is more appropriate for the girl in her situation: accepting the religious engagement and marriage contract and waiting until marriage becomes financially feasible, or postponing all procedures until the young man has the financial means, especially since her family refuses engagement after two previous unsuccessful engagement experiences?

The young man did well by stating that he will not continue an illicit relationship. The relationship between you two is forbidden unless a valid marriage contract is concluded. Therefore, you both must end this relationship. If the young man is of good character and religious, then an attempt should be made to convince your family of him, and to intercede with someone who has influence with them to accept him. If they agree, then all is well and good. Otherwise, you should disregard him and look for someone else. The person whose approval among the family is considered is the guardian (wali). His right is forfeited if his obstruction (adhl) is proven by rejecting suitable suitors without a legitimate شرعي excuse, and the right then transfers to the judge (qadi).