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If the corner of the slit is a sharp corner, that is a stress singularity. That means the stress in the model is a strong function of element size. If the model is remeshed and the element edge length is cut in half, the stress will approximately double, even though there was no change in geometry, loads or supports! That means as you changed slit widths and remeshed, it is likely that the element edge length was varying at the corner so most of the change in stress could be coming from changes in element edge length at the corner and little from the change in slit width.
Make the ends of the slit a semicircle and ensure there are many elements around that semicircle, say 12 elements or more. The stress singularity would be removed from the model and the change in stress when the slit width is changed would be almost entirely due to the slit width with almost no stress change due to remeshing.