The best way to transport her would be with a litter box. This will really help to calm her nerves, and most buns are oriented by food, so just shake a treat bag and she’s likely to go in so she can be transported. Bunnies need a lot of support when being carried so they don’t break their spines, so I’d really avoid doing it unless it’s an absolute emergency. The act of repeatedly picking her up when she’s scared will teach her to fear you, so it really is best to just not do it.

Try to build a bond with her by just sitting with her and ignoring her. Try not to touch her or pet her. Eventually her natural curiosity will take over. You can ask if she’d like to be pet by extending your hand palm down and letting her sniff it. If she lowers her head, you’re all clear to give a few pets. Build up slowly and only do a little at a time so she doesn’t feel overwhelmed. Another great way to bond with her is to hand feed her greens and her pellets.

Unfortunately, yes, that can injure a bun (especially a young bun) – both her bones/joints as well as her internal organs (the liver is particularly vulnerable during those types of holds/grabs). I understand buns are wriggly and need to be handled for practical reasons, but I would suggest trying to lure her with a small treat of pellets into a carrier and transporting her from wherever she is back to her pen that way. Or if the pen is large enough to provide adequate exercise space, leave her there until she gets used to you, her new home, and is ready to start being gradually handled by you with more trust and less fear.

Bunnies are very food motivated and very routine animals in general, so you can limit her feeding area to her pen, meaning whenever you feed her daily pellets or vegetables, do it in the pen only – this can help her associate being in the pen with being fed and after some time she will most likely go back there automatically once she hears you bringing the pellets or vegs.

Note that even after gaining the trust of a bunny, most of them still don’t like being picked up, which is where a carrier comes in handy. Bunnies would need a carrier for their vet visits etc anyway so I suggest getting one and leaving it in the pen with the carrier door always open. You can put some pellets inside as well so they associate the carrier with something good too, instead of just a tool that lifts them up in the air where it feels unsafe.