Just juice is not going to soften up the poop.
Try Prune juice. Just a couple ounces, in his formula bottle.
Or, WHOLE apple juice... the kind that is "murky" looking.
Or, will he perhaps "drink" liquids if perhaps you freeze the juice first... then with a fork, slush it up... to make it slushy. Then, feed it to him with a spoon.
YOu can do this with water too... and then add a little juice to it...
Don't "freeze" the liquids until hard frozen... just until it is semi-frozen, but still can be broken up and mashed into a slush consistency.
Or, give him some Pedialyte. These come in "ice-pop" forms too, to freeze. But at 10 months old, I'm not sure he can eat ice-pops by himself.
Or, make homemade "broth" soups for him, and let him drink this.
The "ideal" poop consistency, Per our Pediatric Gastroenterologist, when my daughter was constipated, is that the poop should look like and have the consistency of cooked oatmeal.
Also, if he can't gum food yet & has no teeth... then don't give him foods that he can't chew. Which he seemingly can't, since he is choking. He's not ready yet for that. Perhaps, this is also why he has constipation... he is not ready for whole foods yet... still puree it, and cook things until it is very soft. No hard chunked foods either, or raw. He will choke with these.
Good luck,
Susan