Summer Internships for 1-Year Masters in Management?

Hi everyone, 

I could really use some advice given my unconventional situation. I did my bachelor's in Hong Kong at a non-target in Hotel Management and graduated June 2020 (midst of the pandemic). Since then, I've been working for almost 1 year at a small VC firm that invests across series A to pre-IPO companies globally and also had the chance to work on the IPO of a SPAC. I will also be taking CFA Level II in November. 

At some point, I realized I really wanted to re-brand myself with a better school name and give myself another shot at entering BB banks or MBB, or at the very least have a higher earning potential. So I applied for HKU Masters in Finance and Masters in Global Management, which I only got an offer to the Management programme and I accepted as I figured I can leverage the HKU Business School alumni network and brand. Coincidentally, I got an offer for an Equity Research role at a small securities firm around the same time. After asking around and overthinking, I ended up deciding to pursue the masters as I really want to work at a large firm and I felt this was the best way. Do you guys think this was the right choice? 

I want to intern part-time while studying if possible and plan on signing up for a career coaching program. I've also networked with a few analysts working at GS. Do I have a chance at all to break into MBB or BB banks (PWM, IBD, or S&T) through a summer internship or am I just wasting my time?

For people who don't know, HKU is the highest-ranked university in Hong Kong and third in Asia. I know BBs in HK historically have hired more overseas from Ivy Leagues, but that has changed in recent years as more are hiring locally from HKU, HKUST, and CUHK from my understanding.