i'm hypermobile too, and always thought I couldn't run, but I have recently managed to - I had insoles made by a specialist hypermobility podiatrist that helped somewhat; then I had others custom-made by Enertor insoles that helped even more for my particular difficulties. I'd try one or either of those. And I had expensive running shoes (Asics) recommended by the podiatrist (I bought three or four pairs recommended by the running shop, brought them in to her, and she watched me run in them with the orthotic insoles, and then chose the best ones and I returned the others). I replaced the running shoes much more often than I expected to - they looked fine on the outside, but the foam does deteriorate over time, so despite hardly wearing them for a couple of years, I bought new ones shortly after I started up again and it helped a lot.
I also tried compression socks, expensive knee brace that worked much better than the cheaper ones, ice pads that go around shins, various ankle supports at times, etc. And started out really really slowly - couch to 5K type thing but a lot more gradual, kind of made up my own routine. And eventually, I managed it despite wobbly joints. I'm still very slow.
It's helped my fitness a lot - I use an Apple Watch, and my 'cardio fitness' score went up steadily for a while. It has now levelled off, which is frustrating, even though I'm running for longer. I also do a little dumbbells arms workout on other days.
I didn't go totally vegetarian, but reduced red meat, and generally only eat meat (usually chicken) a couple of times a week. Sometimes salmon. I tried making a lot of curry/stew/casserole type dishes, adding extra veg, and then making several more portions out of it than the recipe called for, so it was about 2/3 of a normal portion size and more veg in it. Lots of berries rather than other fruit. Reducing cheese to a very small piece once a day with some veg for lunch.
I do still drink some (G&Ts, but more tonic than anything), but never been a big drinker.
Reducing salt and upping potassium. I don't eat many bananas because of the carbs, but I do when it's a longer run that 5K. I also like the small serving bags of popcorn as a treat, only 65 calories, some salt but not a lot, as I sweat a lot of salt out. I eat nuts as a snack as well, though they are quite calorific so I'm trying to cut them down too.
Can't deny that I'm hungry a lot, and crave all the nice things that I've cut out, but my heart is healthier and I feel fitter, and I've bought smaller clothes. I still have high cholestrol (the bad sort) despite all that, and elevated blood pressure, though not enough to need medication, and fairly high blood glucose levels, so i am frustrated, but continuing on this plan for the moment even though I seem to have stalled.
Good luck.