Explanation

 "I would die rather than ask a loan."

After "rather than", we use the base form of the verb (bare infinitive)

When it follows another verb in the same clause, especially after modal verbs like would.

So:

• "I would die rather than ask a loan" (Correct)

• "To ask"  (Infinitive form is incorrect here)

• "Have asked"  (Perfect form is not suitable)