Others have handled the car hitch question quite well. There are a number of places to have one installed if you do a Google search for your area.
A 3 bike hanging rack (trunk style) will be one hot mess. It is a pain to install/uninstall and to make sure the bikes are secure and they won't scratch/rub the other bikes on the rack. The different bike configurations also make it difficult to hang them properly.
I would definitely consider a hitch rack. There are many good options out there. If you are looking to carry 3 bikes, you will most likely need a 2" hitch which will accommodate the bike racks needed for the weight of the three bikes. If you have an e-bike (battery), you will need to pay close attention to the carrying capacity of the bike rack.
etrailer has a great system to walk you through the type of bike rack you might need. You enter the size of hitch, number of bikes, weight of the bikes, etc. and it will give you some options.
Personally, I have a Kuat tray style bike rack which can accommodate up to four bikes and folds down and away to allow access through the hatch back. It handles e-bikes, fat tire bikes and about anything else you can throw at it. It is also the quickest when mounting the bike to the rack (one swing arm and one strap). It is pricey but there are other options. They occasionally go on sale and I got a 20% discount from REI this past spring.
Good luck with your search.