The answer seems easy, don't do the washing up
Srsly, I know this is hard with needles in your legs but try to ignore him. If he gets attention to him it's almost like you're rewarding him.
Reward his good behaviour. Maybe shut him out if poss while you're doing anything that's likely to start him off but allocate time just for bonding and tummy tickles.
A new scratching post and plenty of catnip might help too.
Cutting claws is fairly easy to master, just be careful not to cut the quick (the pink bit) as it can be pretty painful for them. You could get your vet to show you the first couple of times. Mine only charged me £7 and told me how to do it and what clippers to buy.