No, that’s not how rage works.
Murdering random rabbits does not fuel rage. A barbarian’s rage works as follows:
It ends early if you are knocked unconscious or if your turn ends and you haven’t attacked a hostile creature since your last turn or taken damage since then.
First of all, if you’re carrying rabbits around and treating them nicely, then the rabbit presumably is not hostile. It may become hostile if the barbarian suddenly attacks it, but by then it will be dead.
Second, this will require the serial killing of many, many rabbits. Eventually you will run out.
By comparison, it would be more efficient for the barbarian to maintain the rage by damaging themselves. This would fulfill the “or taken damage since then” clause. On rounds where they can’t attack an enemy, they smack themselves with an unarmed attack, or use the hilt of their weapon as an improvised weapon. They have resistance to physical damage during a rage, so this probably won’t hurt them very much.
Recommendation: Make attack rolls anyway.
In general, the barbarian should be attacking every round in a rage. If you really need to keep the rage going, then make attack rolls. They don’t have to hit.
For example, the barbarian could carry some spare throwing weapons or some ranged weapon. That way, if your enemies are out of your movement and melee range, then the barbarian can make a thrown or ranged attack each round while moving toward the enemies. Even if these attacks miss the enemy, they will keep the rage going until the barbarian has closed in.