That should have worked unless you've damaged the seat by scraping it and the reseating tool cant take the damage out. What I would do is swap over the hot and cold mechanism and see which one drips, that will tell you if the fault is the tap seat or the valve. If the latter then I would be tempted to buy a couple of new valves, they aren't expensive but you'd need to take it with you to a plumbers merchant or have it in front of you with measurements if buying on line. There are companies that sell nearly all variations and show diagrams and dimensions.
If the same side still drips after swapping them over then it's the tap seat and you'll just have to keep trying the tool and if no good then it's a new tap, it's unusual but quite possible for the seat to have a hairline crack.
Best of luck, Belfast sinks are damned awful things when anything goes wrong.