If you want to become an engineer then I'd pursue a standard bachelor's program. MS programs are largely fluff, and rushing through an accelerated undergrad is a mistake. Undergrad is not only where you learn the fundamental sciences like kinematics, material science, thermo, etc; its where you start to learn the practical engineering skills needed to succeed - CAD, FEA, CFD, analytical approaches, design of experiments/testing, drafting/GD&T, manufacturing processes/DFMA, engineering quality, PM, etc. Learning the science fundamentals is a fairly quick & easy process, grab a textbook and spend a few evenings reading. Developing the practical skills OTOH takes a few years of hands-on experience. Many juniors are quickly forced out of engineering bc they learned the sciences but not the skills in college (bc many schools put little/no emphasis on the skills) and didn't do the usual 2-3 years of internship to develop reasonably professional competency before graduation.
Regarding degrees, I've prob worked with a half dozen engineers with a non-engineering degree and never doubted their competency. I also had one as an intern for a few years while he finished a BS-Physics and completed a MS-Physics. After graduation he was hired into another dept ahead of many applicants with engineering degrees. To say he worked his backside off for several years playing "catchup" is an understatement tho. There's a lot of crossover between Physics and the fundamentals of mechanics/statics/kinematics/dynamics/etc so he read my textbook library to fill in the blanks, took a few of the "heavier" courses like material science and FEA in-person, and used our corporate training accounts to learn CAD, GD&T, PM, and other skill basics online.
Hiring decisions are based more on your resume than degree bc as mentioned, the knowledge and experience of recent engineering grads varies a LOT. Learning to apply the science fundamentals to a specific industry/niche is a very steep learning curve during your first few years post-grad. To succeed in that, you need to know the fundamental science AND have the skills. If I'm not confident that I can provide a junior engineer basic guidance and have them remodel a part, rerun analysis, retest, and approve a redrawn print in the first few weeks of employment then I'm not hiring them. A $20/hr intern has vastly different expectations than a $80k junior engineer, even tho both are in similar training programs.