IB/PE Careers London Profile Evaluation/Msc Finance Decision

I am a 2025 graduate in Economics from a top 5 UK university with a first class. International student so sponsorship has always been a challenge in recruiting. I have 2 IB summer internships at middle market banks in London (think the likes of SocGen, HSBC, Wells Fargo, RBC etc). I also have 1-2 other relevant finance work ex in my home country at decent brands. Unfortunately, the latter of my two internships chose to gave no FTs that year. I had 2 grad offers in London in Portfolio management and insurance - didnt think the roles would do much so returned my home country to work as an Investor Relations PE analyst at a top 5 financial institution in my country. Still want to work in IB/PE in london, currently considering a Msc Finance route for the same? Is the MSc worth it or should I continue to blindly keep applying and hope that 1/400 applications would give a decent sponsored role? The unis I am considering are LSE, Oxford, HEC, Imperial and Bocconi. Maybe LBS. Might not be able to apply to Oxford this year due to short timeline and high expected avergae on the GMAT. What is my best option to get a London based IB/PE role? Also as my second choice at LSE , is MSc Finance +PE better or Msc Finance and Accounting?