Is it permissible to ship with companies that pay large sums to customs or use "wasta" (mediation/nepotism) to bring products into Egypt, knowing that this is the only available method for shipping?

The ruling on your transaction with the shipping company depends on the ruling concerning customs duties. If customs duties are permissible because they are in exchange for services or due to the state's need, then it is not allowed to evade them, nor is it permissible to ship with a company that devises stratagems to avoid them. However, if the state takes customs duties unlawfully, then it is permissible to use stratagems to evade paying them by dealing with the aforementioned company. You should refer to trustworthy scholars in your country to assess the situation.