I don't think anyone will advise you to do that.
The wheel is designed to fit 120 section tyres and you'd struggle to fit anything much wider in there anyway. You wouldn't improve the handling even if you did, it would (I expect, no experience of it myself) feel heavier and steer slower. All in all not a great idea imo
There are those who think the Dunlops you're on don't inspire confidence, and there are tyres that look "chunkier" just because they have a different tread pattern. So consider swapping both tyres, (although I'd say the Dunlops are alright personally, it's not a night-and-day difference if you go for something better), but I wouldn't go for anything more extreme.