We tow our Camry XLE with push button start (we really hate not having a key) on our Acme dolly.
Our Owner's Manual describes the process, but in short you press the button once without having your foot on the brake. That sets up Accessories and allows the steering wheel to unlock.
However, if you tow for several hours, your battery will discharge. To get around that, you either pull a specific fuse (Owner's Manual) or disconnect the battery. To do that, you can install a fuse isolation/disconnect switch kit, but we find it easier, and less expensive to install a disconnect directly on the battery
Simple, fool proof and a couple of dollars.
Remember to release the Emergency Brake before you drive off!
Fair Winds and Following Seas