Is it permissible to deceive an Islamic bank by colluding with a third party to obtain liquidity for completing construction, knowing that this is the only available means to avoid dealing with interest-based (usurious) banks?

What you mentioned about an agreement with your cousin to buy materials from him in front of the bank, then the bank pays him the amount, and you receive it from him afterwards and return his goods to him, is impermissible. This is because your intention is to obtain a loan with usurious interest, which is known as "money for money with a silk cloth in between" (an idiom for a usurious transaction disguised as a legitimate one). If you are in need of money to complete the construction, you can obtain it through the permissible tawarruq (monetization) by having the bank buy you materials suitable for construction or other purposes, and then you sell them to obtain the money.