We are happy to help you find a hitch rack for your family. The first thing you will want to double check is the size of the hitch on your car. The most common will be a 2in hitch, but there are a lot of 1.25in hitches still out there. It will be important for you to know this as you evaluate your options.
To allow your hatch the ability to open with bikes still on the car, you are going to want to look for a swing away style hitch rack. This option from Thule and this option from Yakima would work really well if you have a 2in hitch. If you have a 1.25in hitch receiver, then unfortunately the swing away style rack is not available. However plenty of 1.25 racks will hold four bikes and still tilt out of the way unloaded. Here is an example of what I am talking about.
All of the racks I have linked to come with security features to protect your bikes and have a four bike capacity.
Feel free to let us know if you have any other questions or want further advice!