I'm non-CS, FCH in Engineering (non-computing related). Gotten an offer from FAANG, but only after 1+ YoE doing tech in another company. So it IS possible.

Alternatives wise, DBS SEED is 5.5k off the bat, but note it's not a permanent position. IME, focus on getting the right skills, have a plan where you want to go, and never stop upskilling. I made the poor choice of not selecting CS for my undergrad (I thought it was a passing fad), but I'm sure you will find your path.

It's true that nobody really cares about the FCH, but you need to tell yourself that it is a testament to your ability to learn, which is most important in your future growth.