Oh my, what an ordeal!
First step, take a deep breath! And maybe another two or three after that. To answer your original question, yes bunnies can die from fear, but usually this is from prolonged fear (such as a dog trying to get into their cage for so long that it gives them a heart attack).
Next, I think the vet is the solution for you (quite a few members go to the vet for trims, especially if they don’t have helpers). But it is important that either your bun is trained to go into his carrier, OR you can confidently pick him up when needed (imagine if there was a fire or other emergency!).
Many members have their buns trained to go into their carriers. It doesn’t need to be a large carrier (in fact too big a carrier is dangerous for the bun in the car), but it should have a side door so they can hop in on their own.
Start by just leaving the carrier out all the time, with the door open. The bun will start viewing it as a hiding spot. You can feed veggies and pellets in the carrier, so they learn to run in for food.
In terms of handling squirmy buns, it is important to be FIRM and quick. If you try to pick him up and fail, wait a bit and try again. The more you chase them around the more worked up they will get.
My Bunston HATED being picked up and was very squirmy, and impossible to trance. If they feel like they can get away or might be dropped, they will squirm and struggle. For nail trims, I always had a helper. I would wear a thick sweatshirt, and hold him pinned against me. He would try to bite and scratch at me, but the sweatshirt would protect me.
For picking up in general, you want to do it in such a way that they are very supported, and also very sure that they won’t be able to squirm away. Firmness is key (obviously not squeezing them too much though!).
I’m right handed, I’ll try to explain my strategy! My technique is to first place the back of my right hand on the ground in front of them (so I’m facing the bunny head-on), as though I’m going to pet them. Then I place my left hand on their bum, so my left forearm is along the left side of them. I simultaneously slide my right hand under their front end on the right side (like their arm pits). Lift the front and and scoop their bum towards me, so their bum is in my left hand, but back feet are not touching anything. So now they are lifted off the ground, with both front and hind end supported. I then immediately pull them close to my body, and usually will tuck their head under my left arm, so their right side is now pressed against my body. Doing this blocks their vision and makes them much less likely to struggle as they are held on all sides. I of course make sure their nose is exposed!
I’ve been able to hold squirmy buns briefly with just my left hand this way (head tucked under my arm, bum in left hand, left side against left forearm, right side against my body), leaving my right hand free to open carrier doors, etc.
I am also tend to do grooming and nail trims on my lap in a chair, or sitting on a towel on a table. Sometimes the table is necessary for a deep grooming during molting. Tough love is necessary sometimes!
And most importantly, keep taking a deep breaths! If you are calm, bun will be more calm.
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The answers provided in this discussion are for general guideline purposes only. The information is not intended to diagnose or treat your pet. Seek the advice of your veterinarian or a qualified behaviorist.