If the places you want to work at require higher technical skills, then getting a masters degree won’t help you achieve that. I don’t know why you would even consider going for your master’s at this point. Give 2 years of slave labor and take on massive debt just to get your name on a paper no one will ever read and stick your toe into various disciplines you’ll probably never revisit.
If you’re really set on data science then you will need a PhD to be competitive in that field. There are a small number of seats with many highly educated people trying to sit in them.
Don’t listen to the recruiter about why you didn’t get the engineering job. Recruiters are only useful for getting short-listed. They usually aren’t part of the hiring committee. The team probably just chose someone else with more working experience for less pay. You can never predict whether you’ll get an offer based on how you felt the interview went because you don’t know who else is interviewing and what rate they were submitted for.
Sorry if this advice sounds like a lot of “Don’ts.” The reason is because you’re doing mostly the right thing — hitting the pavement and maximizing your options. Keep doing that and eventually you’ll get an offer, ideally 2.