Hard to say what you mean by sputters. But it could be something else. In particular, you might have a very bad connection to the motor on the main phase wires.
Once I had a cut wire, that I just kept failing to see, where the wire rubbed a tire. I had two good wires, but the third motor wire was cut down to just two strands big as a thread.
Unloaded, the motor worked great. only needed a fraction of an amp then. But ask it to pull 15 amps, and those two strands could not do that. so then it ran on only two phases, doing what I call motor stutter. Sometimes a motor will run fine on just two phases, but don't try to ride it under load. chunka chunka chunka, and it barely rolls at all.
Since yours runs fine when the battery is full, ( or does it now?) your problem may be entirely battery related. It could be a broken wire to the bms, just out of balance, or even a cell died. But look a good hard look at all your wiring too. 95% of all e bike problems are a plug or a wire not connected right. This could include the battery wire, causing a voltage drop to your system, which then shuts itself off. ( controller low voltage cutoff, or your CA can be doing it if you have it set high)
To fill and balance your battery, charge full, unplug, let sit half an hour. then replug, and charge more. if it won't restart, ride around the block and then recharge. When the green on the charger light turns on, your bms will be discharging any cells that are slightly overcharged, then in half an hour, it can then restart and fill the low cells a bit more. It could take many repeats to get the battery fully balanced. Its balanced when the battery holds full voltage, or very close to it, after you unplug and wait half an hour. If your pack can't hold 58v or more, and keeps dropping to lower than 58v with no improvement each try, then your battery has an actual problem. Ideally, a new battery would balance well, and hold 58.8v for at least a few hours. In reality, it might be full as it gets at less, but not less than 58v.