This is a companion piece to the trailer hitch ratings post I made a few days ago.
Given that the CR-V has a rear-opening hatch, choices in racks are limited.
In the past, before I owned a CR-V, the bike rack I owned was sold by Trek, and would strap to your trunk lid. The key to this rack was that the bottom bar rested on the top of the bumper, so the weight of the bikes was not directly supported by the trunk lid. However, the straps that hooked around the trunk lid (four of them–two towards the front, two towards the side), while cushioned, still could exert force on the metal, and on some cars I’m sure the alignment of the trunk lid could be thrown off or in extreme cases, warpage could occur.
There are still some racks that can hang from the rear of an SUV or similar vehicle, but I do not like those for the same reason, and many no longer rest on the rear bumper. Not only that, the bikes are higher up and will block your rearward view.
Other than the 2021 CR-V Hybrid, all CR-Vs have trailer hitch mounting options. You need not go with a high-capacity hitch, as the CR-V can only tow 1,500 pounds and support a 150 pound tongue weight. I went with a Class III hitch only to get the 2-inch receiver, figuring that I could handle any accessory I mounted to it.
This begs the question: How many bikes can I carry? For a CR-V, realistically you could haul two bikes safely. You not only have to include the weight of the bikes, you also need to include the weight of the bike rack. Better racks that you load bikes on top of (like my Thule 990XT Doubletrack) will be heavier. More on those below. If you are hauling carbon fiber bikes, where the total weight is 20 pounds or under per bike, you could easily haul four. But if you want to move your e-bike around or have old steel framed bikes, then they will weigh substantially more. Some e-bikes exceed 50 pounds, in which case you’d be very close to the 150 pound tongue weight limit.
As for my preferred type of bike rack, I prefer those that cradle the tires and clamp on from the top, to hold them in place. The hanging style racks can allow bikes to swing back and forth and hit each other. (In theory, they are designed to prevent that, but in practice that is not always the case.) Many racks have a feature that lets you swing the track out of the way so you can open your hatch or trunk, and also fold up to the rear of the vehicle when you are not hauling bikes.