Cream of tartar is an acid. The main purpose in a recipe I believe is to work with baking soda to make carbon dioxide to make the dough rise. Baking powder is a mixture of baking soda and various acidic reagents (including cream of tartar) so that some of the reactions are delayed (heat activated) and the gas is not released all at once. If you put baking powder in water, you will see it fizz up somewhat, but once that has died down and if you heat it you should see more fizzing up. The ratio of baking soda to cream of tartar can be calculated fairly easily, and the same can be done if you include baking powder additionally.
The crunch comes from caramelization of sugars which begins at a little above 350F/180C (not to be confused with Maillard browning which begins at a much lower temperature). Now, I have not done any research on this to date and I can only share with you my educated guess. I think if anything, cream of tartar may actually inhibit the crunch as carameliaztion is promoted by alkaline conditions and baking soda is mildly alkaline.
One thing perhaps worth experimenting is to brush the dough with a baking soda solution just before baking. You will need to watch them because you also run the risk of the cookies going brown too quickly before the inside is cooked or they get too dark/burnt. You can also use a solution of sugar and baking soda instead of just baking soda for this "glaze". As for concentrations, you will need to play with that.