Is it permissible for me to request a certificate of weakness (indigence) to obtain an exemption from hospital delivery and 'Aqiqah expenses, knowing that I am in debt, or is it permissible for me to borrow to cover these expenses?

If the certificate of weakness (poverty) has specific conditions, then these must be adhered to, based on the hadith (definition): "Muslims are bound by their conditions." If the conditions do not apply to the questioner, then it is not permissible for him to request it. Instead, he should seek a permissible means to repay his debts and fulfill his needs, such as a benevolent loan (Qard Hasan (definition)). There is no harm in taking a loan for the Aqiqah (definition) (sacrifice for a newborn). It was reported that Imam Ahmad said concerning one who borrowed for the Aqiqah: "I hoped that Allah would compensate him for reviving the Sunnah (definition)."