Fingerless welding gloves are your friends! :lol:
So far, out of several hundred rabbits over the years, I have only had TWO that actively enjoyed being picked up and held. And one of those had a broken back (from before we got her) so, perhaps, she just couldnt kick.
Out of all those rabbits though, only a few dozen didnt like being PETTED! Right now, all but one in my herd of 30 like being petted and she gets a pass through litter size and the fact she isnt actively aggressive, just antisocial.
It does sound like you are too cramped, big rabbits need big space adn all wire cages are the best for odor control. Well, the best I have found anyway. All my cages are set up in deep shade, about 3 feet off the ground adn the manure just builds up into a pile, mixed with spilt hay, and, when I need it for the garden I just fill up my cart from the biggest pile. All sorts of things live in the piles, breaking it down into compost, so after awhile I have no smell problems at all.
If you go with the "let the manure pile up" method try to get solder fly larva to put under your cages, they help break the manure down faster and they eat other fly larva so no barn fly problem.
When I started with rabbits I was actively afraid of them
so, if you stick with them you might also get galloping rabbitosis too.