Are you absolutely certain that the "grittiness" is caused by the cheese not melting, and not because the sauce is curdling? If you cook it too long or too fast, that is what will happen.
If you must use the Kraft stuff (personally, I think it has no flavour compared to real Reggiano), try melting the cheese on low heat in a very small amount of cream first, before you add it to the main sauce pan. If it's still gritty, either it's curdling or you need to use a better cheese.
Emulsions aren't necessarily all about oil vs. water. Alfredo sauce is actually an emulsion of cream and butter, both of which contain varying amounts of both water and fat, and in many cases, if you bought them from a supermarket rather than a farm, also a fair amount of emulsifiers.
Any emulsion is going to be temperamental and not respond well to sudden changes in dispersion. The most important thing to do with one is incorporate new ingredients slowly! If you just dump in a bunch of watery tomatoes, or anything else with enough liquid (water or fat), it's almost certainly going to separate.
Even if you incorporate very slowly and thoroughly, if you upset the balance too much, it might still break. There's no way to know the exact amount you can add without experimenting, unless somebody else has already documented it (not likely).
Sometimes, if your emulsion just creams (see my related answer about mayonnaise), you can restore it to its former glory with sufficient agitation. If it's actually broken then you're in trouble.
Anyway, my advice to you would be - if you want a rosé sauce, then make a rosé sauce, don't waste a lot of time and perfectly good Reggiano cheese trying to start from an Alfredo recipe. I've made a great many tomato and/or pesto cream sauces and the general rule with these (including Alfredo) is that you always start with any oil and vegetables (garlic, tomatoes/paste, etc.), then add your seasonings, then add the cream and slowly reduce it. You don't need or want butter at this point, its flavour will be completely overwhelmed by the other ingredients and it therefore just adds instability.
You might also want to consider using sun-dried tomatoes for a stabler and probably tastier result; they essentially classify as a solid as far as emulsions are concerned, so it's not much different from incorporating pepper or dried herbs. You could just make a regular pesto cream sauce and whisk in some sun-dried tomatoes near the end.
Best Answer
Please do see the excellent answer to the Cacio e Pepe question, which applies to your future efforts.
The recipe you link to is just bad, which is why it's not working for you. Boiling the butter and cheese in the pasta water is wrong, and will always result in clumps of stringy cheese. So the answer is to not use that recipe.
The tradition for all emulsified cheese sauce pasta involves tossing the hot pasta in a bowl with the cheese and a little pasta water. Fettucine Alfredo is no exception to this; it just uses butter as well as the cheese. In all emulsified cheese recipes, it's critically important that the cheese not get above 180F/80C, or you will get stringy clumps like you did.
Some examples of recipes that do work, and show this principle:
Then go and read the other answer linked above, because all of those tips for cacio e pepe also apply here.