Couple thoughts.
1. STRETCH! Don't overwork your fingers and wrist. Give them a break. Stretch. And don't overdo it. Good stretches and regular playing will get you there, it just takes time.
2. "The right way" isn't necessarily the way that works for you. If your fighting tendonitis and muscle issues, "the right way" just might not work for you. Might not work for you anyway, just due to your own personal physiology. Do what works for you!
It took me a long time to master Bb (same shape as the B you're talking about), but now that I have, I love it! Very easy way to slide from Bb to B, C and beyond. BUT, I don't barre that shape. (I just checked and found that I "can", didn't used to be able to) What I do instead is index finger on E, middle finger on A, ring finger on C and pinky on G. I used to have to really THINK about getting my fingers into that position, but with lots of practice, it's natural now and it works for me!
There are a lot of things that a lot of people do in non-standard ways. Reaching around and grabbing the G string with the thumb is supposed to be bad form on uke... it seems that people who have guitar experience do it all the time! Hooking the neck in the crook of your thumb rather than putting the thumb behind the neck is supposed to be "the way". But, my fingers like doing it the wrong way, and it works for me.
Don't let "proper technique" get in the way of having fun! Do what works for you. And as your muscles and tendons get used to playing uke, you can always go back and try what was difficult or impossible before.
And if you flat out CAN'T play a particular chord, find an alternate!