Shimura, I think it is neither shameless nor foolish to continue pursuing being a pilot, especially for a major international carrier. Isn't that what drives us all to pursue this career anyway? I know people who have been working hard at getting into SIA and have spent many years racking up instructor hours, regional airline and GA hours elsewhere before finally making their major airline break. It's obvious if you've gone on your own to pursue a CPL/IR that you love flying and if that's what you love doing, the journey to getting to a major carrier will be rewarding.
I've heard that there are first officer training programs available in America. I dont have any explicit details on such programs but I know they exist. You may want to run a search on the internet. Basically they don't pay well but at the end of the year-long program, you'll end up with F/O hours on turboprops or jets and an FAA CPL. It might not seem like much right now but I think with some F/O experience, u'll have an edge when it comes to re-applying to the big carriers. And it's only a year-long program. Hopefully one day SIA will begin hiring female pilots like CX or Silk. All the best in your endeavors!