Is it permissible to take out an interest-bearing loan with forged invoices to pay off debts in cases of necessity and the absence of an alternative?

It is impermissible to defraud loan-granting institutions by submitting forged invoices. This is prohibited by Islamic law due to its elements of deception and usury. However, necessities permit the prohibited. A necessity is when a person reaches a state of perishing or near-perishing if they do not consume what is forbidden. A need, however, does not permit what is forbidden.

For necessity to be invoked, the following conditions must be met: it must be existing, there must be no other means to avert it, the extent of the necessity must be observed, and the lesser of two evils should be chosen.

If these controls are met in your situation, then there is no blame upon you for committing the prohibited act to avert the necessity. Indeed, a need (even if it does not reach the level of necessity) may permit such a transaction.