You don't need very light leather for a braid - I used one of Sjors' old reins (4mm thick), made two slits in it so I got three strands, and then braided them. the funny thing is that when you've braided it, it feels like it weighs a lot less, really weird!
That's a trick of our nervous system and brain screening and interpreting.
Something that covers a larger area of our hand, even though it weighs the same as something that covers a smaller area, feels lighter.
More nerve endings sending signals ... spreading the interpretation "weight" out more.
I think about this when I'm on a horses back and why I prefer, for the horse, though not for myself, a tree'd saddle.
My same weight feels far less to a horse when spread over a, say 10inch x 20in, tree that distributes the weight, then would my same weight bearing down on a few square inches of my boney bum on the horse's poor back.
And our hands are even better at this kind of strange reverse discrimination. I mean wouldn't one think that our eyes would fool us ... that the larger the object appears to our vision the heavier we might judge it to feel? A braided leather looks 'bigger' than the same leather unbraided, no?
Can we see a picture of some of your braids, please?