Engineering to Management Consulting? SAT/GMAT Question

I'm a rising senior Civil/Environmental Engineering major at a top public engineering public school (think: Michigan, Cal, Texas, Georgia Tech, Illinois) with a 3.55 GPA (that I can bring up to a 3.65 by the time I graduate)

I recently found out about management consulting and it's something that I think I would really enjoy. Not to mention, the salary is much better in the 2nd year than I can make in my 10th as a Civil Engineer. I have good SAT scores (2250+ and National Merit Finalist) and am a good test taker, but mediocre in every other aspect like ECs, leadership, and work experience.

I was planning on going to grad school for two years to get my masters before going into the structural field next year. I wanted to work a few years before trying to get into a T15 MBA program to pivot into a tech PM role. I'm just not interested in being in Civil Engineering for the rest of my life.

So my question is: Is there a score I can get on the GMAT that will turn some heads and give me a benefit of the doubt (like 760+)? I also see that some companies ask for SAT scores, so will mine make a difference at all? Or are the tests just benchmarks to make sure people are getting a above a certain score?

Is it worth preparing the next few months and taking the GMAT in hopes of getting an interview/offer from MBB/T2? I'm planning on getting an MBA anyway so it makes sense to at least try, and no civil engineering firm will sponsor an MBA, so that's another motivation for me to try and get a consulting job instead.

It's not the end of the world for me if I don't get an interview or anything because I can just continue on the path I'm on, but I thought it wouldn't hurt to ask and see if I can aim for something else and make use of the few months I have before recruiting starts.