Online internship at an U.S boutique, from Europe?

Would it be a good idea, as an European, to cold-call U.S boutiques for an online summer "internship"? There's a 5-hour time difference between NY and London. This means that if a typical workday is 9AM-2AM NY time, I would be able to work 2PM-7AM NY time. That's a 13-hour overlap and I could work when they sleep, which seems to be a competitive advantage.

I need some IB experience to put on my resume. Given the current covid context, my only option is to cold-call boutiques and offer them to do some grunt work for free. I'm not expecting any kind of real, structured online internship like the ones organized by BBs and EBs, I'm thinking more about communicating through emails/Microsoft Teams and completing Powerpoints, compiling data, etc. I'll be happy as long as I get to put the name of an investment bank/boutique on my resume, even if that means doing tasks even worse than what summer analysts usually do.

I live in the UK and have two alternatives: target London boutiques or target U.S ones. Now, I'm a bit reluctant with targeting London boutiques, since my current university is not that good. I'll study for an MSc at a top London uni next year so I'd prefer to wait until then to start networking aggressively.

What's your opinion? U.S, or UK?