Okay, yeah, so it was the one I linked. I would advise swapping the berries for the carrot, 1/4 cup is probably reasonable, I wouldn’t do too much more than that without adjusting the other ingredients upwards. Juice wouldn’t really bind, if you want to leave the honey out I’d say a small bit of flour would do wonders. If you read carefully, the author mentions it used to be made with flour instead of ground oats. Oats are a healthier choice, and I love em personally, but flour forms gluten bonds when it contacts liquid (your berry juice) which really give the best structure of anything in baking, so if you’re concerned about it’s ability to hold itself in one piece swapping the tbsp honey for tbsp flour will be perfect!
Of course the honey does work, like it’s like granola bars, keeps it together but in a sticky floppy “glue” sort of way rather than a firmer structured cake/biscuit sort of way. I am not a bunny nutrition expert, but ultimately it’s carbs and simple sugars either way which should be limited for bunnies, so maybe you should just go with either flour or honey and base that decision on what texture you think your bun will like best. Or perhaps just what you have in the house!