Computer Science undergrad having difficulties getting quant internship- what's next?

Hi everyone! I am well aware that there is plentiful of similar posts on this forum asking about computer science majors who wish to become quants- however, my situation is a little different so I hope you guys can spare me the patience in answering the questions.

I'm a Computer Science major from Singapore in my penultimate year specialising in Artificial Intelligence. I did not know much about QF till I made some QF friends recently and realised it is a very interesting intersection of programming, math and finance. I want to try my hand at a quant internship and applied for tens of firms. However, I'm having terrible results thus far- there is either a visa issue or that they decide not to interview me. To be frank I was expecting a better outcome because my grades are in First Class Hons and I've taken several level 4k modules in AI. Of course, my competitors are really good too but I expected to at least get an interview.

I feel that I'm lacking because I do not have the necessary math/finance modules covered. I'm undecided on how I should approach the coming summer break.

1. Should I settle for a software engineering internship? I'm afraid that I'll get pigeonholed into the 'SE role', and I do not really enjoy SE.
2. Should I instead design my own 'summer learning' by taking the relevant math/finance modules this summer and also pick up trading on my own? If so should I join those paid trading courses?
3. What are the other firms I should be looking at? I've also applied for Google and Facebook for their research/data science team. I understand that sales and trading is also a decent alternative.

Lastly, I am considering the option of doing a Masters/Phd in Financial Engineering but as the costs are pretty high, it's ideal that I can work as a quant (or at least as a researcher/trading analyst) after my Bachelors.

Thank you!