Is taking half the profit from relatives' invested money without their knowledge considered a religious transgression, even though their profit was much higher than what they would have received from the bank?

If money is taken for trade and all profits are for your relatives (ibda‘—a form of gratuitous agency where the capital provider receives all profits), then all profits must be given to them, and it is not permissible to take anything from them unless they willingly give you something. The decisive factor is the agreement: if the trade is a donation (tabarru‘), then you are not entitled to any of the profit, but if the agreement is a Mudarabah (definition) contract, then you are entitled to your agreed-upon share.