Your bredli thinks you are a predator. So, stop acting like one. When you pick him up, scoop him up from the bottom. Predators don't do that. You'll probably have to use a hook at first. Just gently wedge it under him and lift him out. Gloves are a good idea as well. They will not only protect your hands but they'll also mask your heat signature and scent, making you appear less predatory.
I would actually suggest that you don't pull him out of his enclosure at all, and I definitely wouldn't take him out of his hide. Rather, I would open the enclosure and allow your bredli to leave it of his own accord. As curious and courageous as bredlis are noted to be, I don't imagine that you will have to wait long. I think you'll find picking him up when he's out exploring much easier. He'll be less defensive and he'll have fewer things to grab onto. If he does choose not to come out, he has his reasons. Respect that and try again later.
Unfortunately the room isn't set up well for free roaming. He'll quickly find his way to a bunch of cables and it's hell getting him out of there. I might be able to set up a good climbing space just outside the enclosure, but can't really let him roam otherwise.