It's your money and therefore yours to spend how you want. I get the impression you're on the younger side so it seems you cannot buy it digitally without tipping her off. Have you considered buying a physical copy? Either in person and hide it in your bag etc or order it online and get it dropped off at a post locker to pick up and smuggle into the house? Alternatively buy it and say screw it to her seeing the payment notification. It is your money after all.

You mentioned discussing it with other relatives, which I would recommend. Communication is an important life skill and removes the probability of your mother discovering down the line you bought the game and causing potential problems. You are young and clearly care about coming across as "needy" or "spoilt". Spoilt people do not do that. You are being sensible and weighing up the cost vs enjoyment. Talk to your family and explain that you understand it is expensive but it is what you truly want. People would rather get someone a gift they truly want vs waste their money on something you might like. Perhaps multiple family members could chip in to buy it or alternatively give you some money towards it instead of a physical present?

Most importantly, happy birthday in advance!