Okay, perhaps the next step would be to find ways to keep the rabbit feeling secure when you lift them, so they don't panic and scramble. A rabbit that feels insecure will be very difficult to manage, but one that feels well-supported can be easier to handle. Yes, there are just plain skittish rabbits that hate any sort of human contact, but usually if I can do things that make the rabbit feel 'safe', things go much better.
When lifting the rabbit, I find one hand on the 'scruff' at the back of the neck while the other hand slides under the rabbit giving support to the rump works well. As soon as I have the rabbit up and out of the pen, I tuck the head under my arm at the elbow, supporting the rest of the rabbit's weight on that arm, leaving me the other arm free to open/close pens or to give additional support to the rabbit.
My mentor told me that if a rabbit starts scrambling, to simply lean over and support the rabbit with your entire body until it calms (can result in stomach scratches if you don't wear a sturdy shirt and you have big rabbits with strong toenails--she raised Flemish Giants).
I rarely have issues when carrying rabbits with the head tucked under my arm. To check the genitals, the rabbit needs flipped over. If the rabbit is tucked under my right arm, I put my left hand on the scruff, and curl my right hand under their rump (they're already resting on that arm.) Now gently pull the rump to the right while using the left hand to flip the head/upper body to the left. The head is now tucked under the left arm, and the rabbit is upside down, leaving the genitals open to the right. I support the rump on a table/cage, so I can use that hand to check the equipment.
Since the head/upper body is supported under the left arm, and the rump supported on a table, I rarely have squirming issues. The secret is to keep them feeling securely held. If they feel like they're going to fall, they'll squirm like crazy.
Now that you've stretched the area below the anus, you should either see a tube pop up (the penis of a male), or a slit appear (the female). Being able to check this at breeding time is important. While the little kits won't be ready, by six months (eight months for the giant breeds), you want to know whether a doe is likely to breed. A receptive doe generally has a moist reddish-pink to purplish slit. A doe with a pale, dry slit isn't likely to be interested in breeding.
