She could be moody, but that doesn't excuse kicking, IMHO. I don't know the full situation (you describe her as a loan...), but if you are the one dealing with her feet day to day, this should be resolved. My only reason for the qualification is that it takes consistency, and if the primary handler isn't following through, life will be harder for all involved.
I would start with a lunge whip or other "arm extender" and get her comfortable with you touching her everywhere with it, especially around her hind end. Trace her outline, search out the trigger spots and rub them away. Keep touching her until she relaxes, then remove the whip. When the whip doesn't bother her (no kick threats, pinned ears, tension, etc.), and you feel safe, try tracing her again with your hand. If she reacts negatively, go back to the whip and get the relaxed attitude back. Next, try a brush. When she's good for all that, pick up her feet.
If she won't lift them nicely, you can pinch the tendon area at the back of the cannon, or the chestnut. Either of these places will probably get her to lift, but some horses work better with one or the other. If you've done the preliminary stuff, she shouldn't kick, but if she does, try to stay close (reduces force of kicks if she hits you) and hang on. If you let go, she got her release of pressure. Try to stay with her until she relaxes, then drop the foot. Work your way up to holding the hoof up longer and longer, and picking it out while it's up. When she's good for the whole bit, you can set her foot down, but for now it's more important to keep her polite and not leaning.
When she stumbles, is it more in her front end or her back end? It could be the length of her toes, but if she's tender in her back end, that could contribute to her "goosiness" and her stumbling. As always, check with a vet, chiro, etc. and do your best to rule out any physical cause of behavioral problems.
I hope that was a bit helpful, and good luck!
A stubborn horse walks behind you, an impatient one in front of you, but a noble companion walks beside you ~ Unknown