In addition, it isn't all about meat. We eat meat every day, either beef or pork or fish or seafood or poultry - but not for every meal certainly, not huge portions, and not the most expensive meat we can find. I shop for sales and as a rule won't pay more than $1.99/lb for any meat. So when chuck roasts are on sale for $1.69/lb, I buy 10 or 20# of it, cut some of them into steaks, some into small pieces for stir-fries or soups, and freeze it all. If we only eat beef twice a week, that $30 purchase lasts a LONG time. Not expensive at all. Ditto pork and fish. The more expensive seafoods, like shrimp and scallops, we use small portions of in salads or as embellishments to a meal but not as a main dish.
Mainstays in our freezer are ground turkey (69¢/lb - add a spoonful of beef base and it's hard to tell the difference between this and $2/lb ground beef!) and chicken breasts or thighs that I buy in bulk and repackage into smaller portions. We also buy bacon and sausages in bulk (10# for about $1.50/lb); we're still working on the 10# box of thick-sliced bacon that I bought 2 months ago! Other meats are only bought on special, as noted above.
We keep and eat lots of frozen vegetables which I purchase in large bags from Gordon Food Service stores. Spending, on average, $2.50 per 2# bag of frozen broccoli, asparagus, green beans, cauliflower, and brussels sprouts, I buy 3-4 of each and have all of our frozen veggies for the month for under $50 - for the MONTH! And of course we also have fresh veggies and romaine leaves for wraps, which I buy from the local produce stand for bargain prices as well by shopping the day-old racks (for that night's dinner) and sales. Even considering that I spend a little extra to buy bags of slaw mix for stir fries and other dishes instead of heads of cabbage, and the ready-made salad mixes, our greens are only about a buck per 1# package at places like Aldi's and Sav-a-Lot. I also purchase dairy and eggs from one of these sources; prices are comparable among them, so it just depends where I happen to be shopping when we need these items; ditto sugar-free gelatin mixes.
The only other foods I buy are a very few special foods from Netrition, which cost me just about a dollar a day, on average. They have a wonderful product called CarbQuik that I buy several pounds at a time; a 3# bag lasts about 2 months. I also get High Gluten Wheat Flour there, and a few other miscellaneous odds and ends which vary depending on my mood at the time I place my orders. But these 2 are staples, and I use them once or twice a week for biscuits, Fiber Biscuit Rounds, pizza crust, etc.
My husband was always a big pop drinker, and I drink a LOT of coffee. But now we drink the GreatValue sugar-free drink mixes from Walmart. A plastic container containing 6 little cups of mix, each making 2 quarts, is $1.96. Even if he drinks the inexpensive 50¢/2 liter pop, the GV drink mix is still much less expensive by a third! And I've been replacing about half of my coffees with it each day. So we're saving money there as well. And of course, water is the least expensive, and healthiest, drink of all!
Overall, and considering the out-of-pocket things we have to pick up during the week like half-and-half or a dozen eggs, our family of 3 eats for $50-60/week. Even people who don't eat low carb rarely do that! So my answer is, "No, it isn't more expensive to eat low carb."
I've heard that low carbers should shop the walls of the store, and avoid the aisles. I've certainly found that to be the truth! We can find the vast majority of everything we eat in the produce, dairy, and meat departments of most supermarkets.
Still not convinced? Here is a list of the foods I purchase on a typical monthly shopping trip. Two weeks later I go back and fill in the perishables like fresh produce and dairy - but I never spend more than $50 on those, and usually much less!