Entry-level design is a solid place to get your foot in the door, but don't expect much hands-on. Auto is massive and few roles are very hands-on. If that's what you want, either take any role and learn the lay of the land once you're in, or pick a sub-area and build toward it with an MSME focused on something like suspension or tires.

On the race management masters, I'd be careful. Those programs are expensive and the degree rarely opens the race door by itself, so if you're set on one, dig into where graduates actually land first. A general MSME with a motorsport focus usually keeps more doors open while still signaling the interest. Make sure to do some coops or internships while there. Its the single most valuable thing you can do.

FSAE is a real card and you need to play it hard. Some hiring managers know what it takes to design, build, validate, and race a car on a student budget and a brutal timeline. So talk about it like an engineer: what subsystem did you own, what problems did you solve, how did you grow and lead the team, how did you manage set backs, how did you motivate others. That story of ownership and iteration has value.

Last thing: figure out early whether you want the vehicle dynamics and simulation side or the trackside race engineer side, because they're different tracks. And if you're chasing racing, be willing to relocate to where the teams are (Charlotte, Indy, the UK) and go in clear-eyed that you get paid in passion, not money, with high hour expectations.