I am severely hypermobile as well, to the point of having health issues. I also have small hands and my span in the RH is further limited by a joint and tendon problem on the thumb.
When I started to play the piano I tried to follow the instructions and developed a lot of tension problems and carpal tunnel symtoms. Only after I found a way to compromise between the ideal playing technique and what my physique allows I started to feel more comfortable with my playing. Now I understand that some things I cannot practice away. The problem is that teachers don't seem to know how to handle such limitations of physique so you are quite alone in trying to work with them. I've been with my teacher for over 5 years and now she gets that I literally cannot do some things with my hands and fingers, it's not a matter of not trying enough or not wanting to practice. Also all the BS about hand shape and size not mattering can be really demotivating. The piano keyboard is ideal for some hands and less so for others. To admit that is the first step to develope a healthy way of playing despite one's physical limitations.
How it works for us now:
I try to play something the way my teacher advices. If that does not work or I get nasty feelings from practicing that way, I discard it and concentrate on trying to find some other comfortable way to get the music sounding the way it should sound. If my hands look bad doing it who cares if it feels and sounds fine. Finding suitable fingerings is a big part of it.
I too have to keep my nails extermely short to play with curved fingers. I have almost no pads on them so they easily slip and the nail joint gives away. Whenever possible I play with a flatter hand and fingers for better span and to avoid wrist pressure. It's important to learn how to keep the knuckles from collapsing, but at the same time avoid too much tension on the forearms that is caused by trying to keep the finger nail joint from giving away. Despite the times of frustration I think I am fortunate to have a teacher who concentrates heavily on technique and who does not want to give in and just let me play sloppily or weakly. It has forced me to spend countless of hours on the piano just analyzing what I do and if I can make it work better somehow. Watching real pianists play has been somewhat helpful too, not to imitate, but to see that there really are different ways to achieve beautiful sound.
After getting a new baby grand this summer I realize it also makes a big difference what you practice on. I can play with different pianos, but continuous practice with an intrument that is difficult to play with my hands won't make me play better but quite a opposite. Practicing without extra tension has made a huge difference in just a few months.
Finger exercises are useful to some extend, but they have not done anything to "correct" my hands. But exercise in general is really important for me, when my neck, back and shoulders get stiff my playing suffers immediately. So I try to exercise regularly even if I never liked it.
The bolded part would worry me a little:
"In her opinion, it is unhealthy for my neck, fingers, and wrist, and I play with added pressure and extra tension that could be easily relieved with proper technique." If you are very hypermobile and have small hands it will not be easy to work on your technique. So my advice is to work on it, but on your own terms and be frank and firm with your teacher. If you are uncomfortable to do what you are asked for or just don't understand how because your fingers cannot, say it. Forcing your hands may develope problems instead of solve them.