Can you restore cars in the Marine Corps?

I'd recommend waiting until you are established at your duty station and have a better idea of your future. All the major Marine Corps bases and most of the Navy bases where you might be assigned have "Auto Hobby Shops" that are well equipped for your project, but you may be assigned to a unit preparing for a deployment or other operational commitments that would limit your free time, Generally, there are open storage lots available collocated with the shops, but if you've got storage available now, it would be better to check things out before moving your project.

You'll want to find out where you're going first. The USMC only has a handful of locations compared to the Army, but once you know where you're going you can look on their website and FB page. If you're going to Okinawa, there's no way you're getting that thing to Oki and back with your weight limit.

Some good answers here. In addition, I’d research the alternative scenarios if you do bring it to your current duty station; e.g., sudden deployment, sudden change in duty and/or station, end of current tour reassignment. You should be prepared for unexpected changes.
Biggest concern I would have is securing the vehicle and parts for all the above situations.