Affair with a Muslim girl and her family is looking for another alliance

Dear writer, thanks for writing in. It's good that you are aware of what your options are. Since you both are clearly not in favour of eloping are you then keen on marrying each other right away or in good time? Focus on the option that is available, that being confronting her parents. Regardless of their agreeing to this, the only way forward is to let them know. Again, before you even approach her parents you have to be sure that you and your girlfriend are on the same page and are just as committed to each other even when circumstances are tough.
I am assuming your family is in know and approves since you see only her parents as a challenge. It might work to your advantage to rally all the support you do have and probably have your parents talk to them as well. Sit them down and talking to them calmly and respectfully. Communicate what your intentions are that you are respectful and accepting of her culture and background. Communicate what you like about your girlfriend and why you both are suited to each other.
Let her parents see that you are worthy of their daughter and that you can be trusted. Any parent has a right to be concerned about their child's choice of life partner and would want only the best for them. So you might have to work at letting her parents truly believe that you the best partner for their daughter. For this to happen you have to be settled professionally and financially or atleast on your way there since these show that you are responsible, hard working/smart, successful and dependable. Let them know that you equally supportive of professional or personal decisions your girlfriend makes. Invite them to ask you questions they are concerned about as well. Let them know about future plans and how you plan to live your life as an intercaste couple; that your are at ease and accepting of this mingling of religions and that you both will not let religious differences affect your relationship or use it against each other.