I can STRONGLY recommend Cappy's. Automotive Transportation. They are California based, have competitive if not rock bottom pricing, and are easily the best company I have used. I've used them for everything from a '91 Porsche 911 to a '56 MB 300SL Gullwing. Perfection. I think they only do enclosed shipping. Great communication.
For a more "mainstream" company, my favorite is Montway. Most of their rigs are company-owned, not independent operators. It gives them better control over who goes where and when. Finally, Passport is OK. You can contact Camille Davis at(417) 650-0972. They are the "official" shipper for the Gullwing Group rallies, so obviously, they know their stuff, but I have had mixed experiences with timing and communications.
That said, I'd put some sort of tracker in the car. I've used this GPS Tracker - Optimus 3.0 4G LTE Bundle with Waterproof Twin Magnet Case - 1 Month Battery which is great. It only costs about $40, and last pretty long, depending on the settings, but it does require a monthly subscription. You can sign up and cancel after you ship. Alternatively, you can just stick an Apple Airtag if you use IOS, or a Tile (both about $35 on Amazon.) Neither requires a subscription. They are not as quick to report but you don't need a subscription and they last hella long.
I have Airtags in all my cars - classic and contemporary- , and also on my keychains. This is both for theft protection and for finding keys and cars when I misplace them. VERY cheap insurance. I misplaced keys at this huge event and the Findmy app got me to within 10 feet of them.
Optimus