We used my 180 frequently for traveling starting when our son was about two. First, we used a high-backed booster in the backseat. As he grew we moved that to the front seat, and then when he grew out of that we put him in a booster seat in the front seat with the aircraft harness as his restraint. There was an awkward age in there where he was too big for a booster seat but still not tall enough to be protected by the shoulder harness, so he had to go back to the back seat for a while. He's a teen now, and when he flies with me (which isn't very often) he sits in the front seat like any other person.
My wife used to be a certified car seat installer, so she had some strong opinions about putting our son in the 180. Her certification has lapsed, but she offers this advice:
"An appropriate car seat is one that fits your child best based on her weight, height and your vehicle or in this situation plane. She absolutely needs a 5 point harness system and good head and neck protection. (No particular brand but some brands might fit better in your plane.) This of course needs to be installed most likely with a locking clip to your aircraft belt system. Are you planning on just using this seat in your plane? If so, the Recaro high backed booster with a built in harness system might work. It is usually more narrow than some of the bulkier convertible seats. The Britax grow with you seat also tends to be more narrow. We also found that our son’s seat fit better in the front seat because of smaller width seats in the back. The Diono convertible car seat also tends to be more of a chair style seat so it might be a good option. Smaller base and larger at top for protection. Good luck!"
I also bought him a child's headset. He refused to wear that. At the time, I had a Bose X ANR headset for me and used passive Peltor headsets for passengers. Of course, the only one he'd wear was the Bose so I had to buy another one of those too (used, on eBay, of course).
I hope y'all have fun and build a lot of memories! We had some great trips in those days, and the time goes fast.