Your neighbour sounds like a real gem, not. They can send you all the emails they like, but there is an order requiring them to do something, and they haven’t appealed that order.
If it were me, I wouldn’t enter into any correspondence with the owner other than to remind them of the order and that there are penalties for failing to comply with the order of an adjudicator, and that you will pursue the matter if they don’t meet the terms of the order.
The ‘root problem’ is that the underlay is insufficient, as set out in the adjudicator’s orders. Whether installing Regupol or carpet with underlay will be sufficient to solve your issue can probably only be answered by an expert on matters like that. I am not sure if you have consulted anyone on that and it may be a good idea if you do. The Regupol is good, but it won’t completely do away with all noise. So it depends on what you want. Perhaps you need to ask for what you really want after seeking expert advice, and go from there. But don’t let your neighbour dictate the terms, they are the ones with an order against them, not you.