I would suggest using a proper mortice chisel. You'll have to make or grind one thinner. Look up how to use a mortice chisel if you're not familiar with this.
I'll second Jim's suggestion. A mortising chisel has a deep body, with about 30 degree bevel. The deep body gives enough strength that you can avoid breaking off the tip when cutting mortises. And the depth also helps keep the chisel running straight. These chisels have softer steel than carving chisels, so they are resistant to breaking.
(My carving chisels have more shallow bodies and very hard steel with more shallow bevels of 25 degrees or less, and the tips can break if they are driven in too deep or if any sort of leverage is involved. Cutting a blind mortise like a trigger mortise requires a little leverage action as you scrape out the bottom.)
Paul Sellers has a good video showing his technique for cutting mortises. His vid shows two chisels. One is an old style mortising chisel, designed for cutting deep mortises quickly without breaking. It seems to me that the one he is using is overkill for the size hole he is cutting. I think that's the main reason he struggles with it. It's just too big for the hole. The second one he shows is a carpenters' chisel. He uses the same technique in both instances, so he isn't meaning to show two techniques. He is just showing that you can use the carpenters' chisel to cut a mortise. Both have a bevel of about 30 degrees.
Two things Paul does not show. One is he doesn't show cutting-in along the length of the mortise. That's not essential for something that's going to be covered up, but it makes for neater work, as it helps avoid tearing the edge. Second is that he does not show measuring the depth of the holes. He doesn't need to do that in this video, as he's cutting the mortise behind glass, so as to show the technique. But if cutting a real blind mortise, some sort of depth gauge is essential, even if it's just a toothpick.
