Unfortunately there's not too many options from here - i.e. you can either trust that the phone developed these faults on its own and accept a return, or not trust that and refuse (unless the buyer does open a request).
You could ask for a video of these faults occurring, and / or try to help troubleshoot (eg did it start happening after the batter was replaced), though the buyer needs to be willing and able to particpate in that. I'd personally just be honest with them, in situations like this I often will say something like I'm happy to help any way I can, and then ask for evidence of the claimed fault (usually photos, though they won't really help here) and tell them it's so I can get a proper understanding of the problem and determine the best course of action to take. I also add that if they can't provide that, then they can return the item and once received it will be inspected and refunded if found to be faulty, with return postage reimbursed.
You've already partially refunded and they have a phone, plus (presumably) both the original and old battery, but if accepting the return, I'd request that they return with the original battery, as the previous deal with regards to the partial refund to buy a new battery was offered on the understanding they were keeping the phone, and is void if the phone is returned (meaning that's a cost they will have to bear themselves).
