Alyass :
Your current one will not suffice for this build, W would suggest going for a 400W one. There is no way around this.
Update on the situation.
I ended up getting the 750 ti shortly after I posted this. I have used it now for months. This is all on the 250W power supply. I have not had any problems with overheating, noise or sudden shutdown. So it ended up working fine with my weak power supply after all. It didn't require a PCI-E cable either, if that should be of importance.
If I encounter any problems in the future I'll be sure to post them here, but so far it seems like a weak 250W was able to support the card.