Former NAME major here! One thing that I don't think you're fully grasping is that most any career is open to any major. Management majors can be engineering officers, and engineering majors can be deck watch officers. The fastest way that anybody can get to cutter command is to be LTJG and take command of an 87' patrol boat. This would be your second tour. I won't say it's impossible for a Student Engineer (that's what they call first tour officers who work in the engineering department on the ship - again, no direct relation to major), but it is incredibly unlikely. Much more likely is Executive Officer on a Fast Response Cutter or Icebreaking Tug for your second tour. This would help move you onto the path for command. If you're talking major command (210+ foot cutter), neither one is really faster than the other. Being a student engineer from the Academy leaves every job in the Coast Guard open for your second tour, which isn't true of anything else. So, to be perfectly clear: No major will help you get there faster. You should do what you enjoy the most, because that is most likely what you will do the best in, and a high class rank could help you get there faster.
Again, open to many other majors besides NAME, and you don't have to this because you're NAME, but typical shore tours for NAME majors (or anyone in the Marine Engineering Specialty in the Coast Guard) may include working at The Yard, being a port engineer, working for a specific product line, or working at a Project Resource Office overseeing the building of new cutters.
As alluded to above, many stations are commanded by senior enlisted and Warrant. For those that are commanded by O's, you would need to be a minimum of an O-3 LT. Realize that being a station CO and a cutter CO are somewhat competing goals, because station CO's come from the Response Ashore community, while cutter CO's come from the Response Afloat and Prevention Afloat (cutter) communities.