What is the legitimate Sharia-compliant way to finance the completion of a paint factory owned by the father, given the son's refusal of usurious borrowing, the absence of Islamic banks in Syria, the lack of liquidity, the father's illness, his urgent need for work, and his inability to sell a property he owns?

It is not permissible to borrow with interest because it is prohibited usury, except in cases of dire necessity, and there is no necessity in your situation. Your father must repent sincerely, and you must advise him by clarifying the prohibition of usury and that he should be the furthest of people from unlawful acts in his current state.